{"question": "In triangle ABC with coordinates A=(-7, -10), B=(-2, -3), and C=(-3, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "17.10°", "metadata": {"A": ["-7", "-10"], "B": ["-2", "-3"], "C": ["-3", "-6"], "angle_ABC_degrees": 17.10272896905237, "source_dataset": "advanced_geometry", "source_index": 0, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, -6), B=(4, 1), and C=(-7, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.304, -1.217)", "metadata": {"A": ["-1", "-6"], "B": ["4", "1"], "C": ["-7", "4"], "ortho": ["7/23", "-28/23"], "orthocenter_approx": [0.30434782608695654, -1.2173913043478262], "source_dataset": "advanced_geometry", "source_index": 1, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 7), B=(-7, -5), and C=(2, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.176", "metadata": {"A": ["6", "7"], "B": ["-7", "-5"], "C": ["2", "-3"], "incircle_radius_exact": "sqrt(-sqrt(29) + sqrt(85)/2 + sqrt(313)/2)*sqrt(-sqrt(313)/2 + sqrt(85)/2 + sqrt(29))*sqrt(-sqrt(85)/2 + sqrt(29) + sqrt(313)/2)/sqrt(sqrt(85)/2 + sqrt(29) + sqrt(313)/2)", "incircle_radius_approx": 2.176123777286009, "source_dataset": "advanced_geometry", "source_index": 2, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, 5), B=(-2, -8), and C=(-1, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.951", "metadata": {"A": ["3", "5"], "B": ["-2", "-8"], "C": ["-1", "0"], "incircle_radius_exact": "sqrt(-sqrt(194)/2 + sqrt(41)/2 + sqrt(65)/2)*sqrt(-sqrt(41)/2 + sqrt(65)/2 + sqrt(194)/2)*sqrt(-sqrt(65)/2 + sqrt(41)/2 + sqrt(194)/2)/sqrt(sqrt(41)/2 + sqrt(65)/2 + sqrt(194)/2)", "incircle_radius_approx": 0.9509128146769613, "source_dataset": "advanced_geometry", "source_index": 3, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, -9), B=(8, 8), and C=(-3, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.945, 7.196)", "metadata": {"A": ["2", "-9"], "B": ["8", "8"], "C": ["-3", "10"], "ortho": ["984/199", "1432/199"], "orthocenter_approx": [4.944723618090452, 7.1959798994974875], "source_dataset": "advanced_geometry", "source_index": 4, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 3), B=(7, 4), and C=(8, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.721", "metadata": {"A": ["-8", "3"], "B": ["7", "4"], "C": ["8", "0"], "incircle_radius_exact": "sqrt(-sqrt(17)/2 + sqrt(226)/2 + sqrt(265)/2)*sqrt(-sqrt(226)/2 + sqrt(17)/2 + sqrt(265)/2)*sqrt(-sqrt(265)/2 + sqrt(17)/2 + sqrt(226)/2)/sqrt(sqrt(17)/2 + sqrt(226)/2 + sqrt(265)/2)", "incircle_radius_approx": 1.7214510174925386, "source_dataset": "advanced_geometry", "source_index": 5, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, -6), B=(7, 7), and C=(-1, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "54.57°", "metadata": {"A": ["0", "-6"], "B": ["7", "7"], "C": ["-1", "6"], "angle_ABC_degrees": 54.57422788509182, "source_dataset": "advanced_geometry", "source_index": 6, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, 3), B=(-7, 0), and C=(7, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(58.000, -130.000)", "metadata": {"A": ["1", "3"], "B": ["-7", "0"], "C": ["7", "6"], "ortho": ["58", "-130"], "orthocenter_approx": [58.0, -130.0], "source_dataset": "advanced_geometry", "source_index": 7, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, 1), B=(10, -3), and C=(5, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.626", "metadata": {"A": ["-2", "1"], "B": ["10", "-3"], "C": ["5", "0"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + sqrt(34)/2 + 5*sqrt(2)/2)*sqrt(-5*sqrt(2)/2 + sqrt(34)/2 + 2*sqrt(10))*sqrt(-sqrt(34)/2 + 5*sqrt(2)/2 + 2*sqrt(10))/sqrt(sqrt(34)/2 + 5*sqrt(2)/2 + 2*sqrt(10))", "incircle_radius_approx": 0.626195388715472, "source_dataset": "advanced_geometry", "source_index": 8, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 7), B=(-5, -3), and C=(-3, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.000, 9.200)", "metadata": {"A": ["6", "7"], "B": ["-5", "-3"], "C": ["-3", "7"], "ortho": ["-5", "46/5"], "orthocenter_approx": [-5.0, 9.2], "source_dataset": "advanced_geometry", "source_index": 9, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, 6), B=(5, 1), and C=(3, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.388", "metadata": {"A": ["-9", "6"], "B": ["5", "1"], "C": ["3", "-9"], "incircle_radius_exact": "sqrt(-sqrt(221)/2 + sqrt(26) + 3*sqrt(41)/2)*sqrt(-sqrt(26) + sqrt(221)/2 + 3*sqrt(41)/2)*sqrt(-3*sqrt(41)/2 + sqrt(26) + sqrt(221)/2)/sqrt(sqrt(26) + sqrt(221)/2 + 3*sqrt(41)/2)", "incircle_radius_approx": 3.388032708309733, "source_dataset": "advanced_geometry", "source_index": 10, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 4), B=(6, 5), and C=(6, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "84.29°", "metadata": {"A": ["-4", "4"], "B": ["6", "5"], "C": ["6", "1"], "angle_ABC_degrees": 84.28940686250036, "source_dataset": "advanced_geometry", "source_index": 11, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, 7), B=(-1, 5), and C=(5, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.444, 5.889)", "metadata": {"A": ["4", "7"], "B": ["-1", "5"], "C": ["5", "2"], "ortho": ["31/9", "53/9"], "orthocenter_approx": [3.4444444444444446, 5.888888888888889], "source_dataset": "advanced_geometry", "source_index": 12, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, -6), B=(-1, -8), and C=(-5, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-11.000, -10.000)", "metadata": {"A": ["-4", "-6"], "B": ["-1", "-8"], "C": ["-5", "-1"], "ortho": ["-11", "-10"], "orthocenter_approx": [-11.0, -10.0], "source_dataset": "advanced_geometry", "source_index": 13, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, 5), B=(10, 6), and C=(-1, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.06°", "metadata": {"A": ["-10", "5"], "B": ["10", "6"], "C": ["-1", "7"], "angle_ABC_degrees": 8.056834133846554, "source_dataset": "advanced_geometry", "source_index": 14, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, 8), B=(9, -10), and C=(-3, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.273, 7.455)", "metadata": {"A": ["1", "8"], "B": ["9", "-10"], "C": ["-3", "6"], "ortho": ["3/11", "82/11"], "orthocenter_approx": [0.2727272727272727, 7.454545454545454], "source_dataset": "advanced_geometry", "source_index": 15, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, 2), B=(1, -1), and C=(-2, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "75.96°", "metadata": {"A": ["4", "2"], "B": ["1", "-1"], "C": ["-2", "4"], "angle_ABC_degrees": 75.96375653207353, "source_dataset": "advanced_geometry", "source_index": 16, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 10), B=(4, -10), and C=(-6, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-31.870, -16.522)", "metadata": {"A": ["-8", "10"], "B": ["4", "-10"], "C": ["-6", "-1"], "ortho": ["-733/23", "-380/23"], "orthocenter_approx": [-31.869565217391305, -16.52173913043478], "source_dataset": "advanced_geometry", "source_index": 17, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, 8), B=(-6, -2), and C=(-3, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.223", "metadata": {"A": ["-1", "8"], "B": ["-6", "-2"], "C": ["-3", "5"], "incircle_radius_exact": "sqrt(-sqrt(13)/2 + sqrt(58)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + sqrt(13)/2 + sqrt(58)/2)*sqrt(-sqrt(58)/2 + sqrt(13)/2 + 5*sqrt(5)/2)/sqrt(sqrt(13)/2 + sqrt(58)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 0.22319770263487862, "source_dataset": "advanced_geometry", "source_index": 18, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 7), B=(-4, 0), and C=(-1, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.158", "metadata": {"A": ["-5", "7"], "B": ["-4", "0"], "C": ["-1", "0"], "incircle_radius_exact": "sqrt(-3/2 + 5*sqrt(2)/2 + sqrt(65)/2)*sqrt(-5*sqrt(2)/2 + 3/2 + sqrt(65)/2)*sqrt(-sqrt(65)/2 + 3/2 + 5*sqrt(2)/2)/sqrt(3/2 + 5*sqrt(2)/2 + sqrt(65)/2)", "incircle_radius_approx": 1.1580887317290323, "source_dataset": "advanced_geometry", "source_index": 19, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, -8), B=(-3, 4), and C=(-1, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "43.15°", "metadata": {"A": ["-5", "-8"], "B": ["-3", "4"], "C": ["-1", "1"], "angle_ABC_degrees": 43.1523897340054, "source_dataset": "advanced_geometry", "source_index": 20, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, -1), B=(-2, 5), and C=(-8, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.959", "metadata": {"A": ["4", "-1"], "B": ["-2", "5"], "C": ["-8", "2"], "incircle_radius_exact": "sqrt(-3*sqrt(17)/2 + 3*sqrt(5)/2 + 3*sqrt(2))*sqrt(-3*sqrt(2) + 3*sqrt(5)/2 + 3*sqrt(17)/2)*sqrt(-3*sqrt(5)/2 + 3*sqrt(2) + 3*sqrt(17)/2)/sqrt(3*sqrt(5)/2 + 3*sqrt(2) + 3*sqrt(17)/2)", "incircle_radius_approx": 1.959162194043828, "source_dataset": "advanced_geometry", "source_index": 21, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-7, 10), B=(9, 2), and C=(7, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.819", "metadata": {"A": ["-7", "10"], "B": ["9", "2"], "C": ["7", "-10"], "incircle_radius_exact": "sqrt(-sqrt(149) + sqrt(37) + 4*sqrt(5))*sqrt(-sqrt(37) + 4*sqrt(5) + sqrt(149))*sqrt(-4*sqrt(5) + sqrt(37) + sqrt(149))/sqrt(sqrt(37) + 4*sqrt(5) + sqrt(149))", "incircle_radius_approx": 3.818813450082334, "source_dataset": "advanced_geometry", "source_index": 22, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -1), B=(6, -3), and C=(4, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.080", "metadata": {"A": ["-1", "-1"], "B": ["6", "-3"], "C": ["4", "4"], "incircle_radius_exact": "5*sqrt(2)*sqrt(-5*sqrt(2)/2 + sqrt(53))/(2*sqrt(5*sqrt(2)/2 + sqrt(53)))", "incircle_radius_approx": 2.080319990748767, "source_dataset": "advanced_geometry", "source_index": 23, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, 3), B=(-3, 4), and C=(-1, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.000, 4.000)", "metadata": {"A": ["-1", "3"], "B": ["-3", "4"], "C": ["-1", "-2"], "ortho": ["2", "4"], "orthocenter_approx": [2.0, 4.0], "source_dataset": "advanced_geometry", "source_index": 24, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, 3), B=(4, 3), and C=(-2, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.000, 8.000)", "metadata": {"A": ["-7", "3"], "B": ["4", "3"], "C": ["-2", "9"], "ortho": ["-2", "8"], "orthocenter_approx": [-2.0, 8.0], "source_dataset": "advanced_geometry", "source_index": 25, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, 6), B=(-7, 9), and C=(8, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "23.40°", "metadata": {"A": ["4", "6"], "B": ["-7", "9"], "C": ["8", "-3"], "angle_ABC_degrees": 23.404689551032316, "source_dataset": "advanced_geometry", "source_index": 26, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, -7), B=(-5, -8), and C=(9, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "133.23°", "metadata": {"A": ["-9", "-7"], "B": ["-5", "-8"], "C": ["9", "1"], "angle_ABC_degrees": 133.2285302599659, "source_dataset": "advanced_geometry", "source_index": 27, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, 4), B=(4, -6), and C=(0, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.556, -6.222)", "metadata": {"A": ["-1", "4"], "B": ["4", "-6"], "C": ["0", "-7"], "ortho": ["14/9", "-56/9"], "orthocenter_approx": [1.5555555555555556, -6.222222222222222], "source_dataset": "advanced_geometry", "source_index": 28, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 9), B=(-10, -2), and C=(-6, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.828", "metadata": {"A": ["6", "9"], "B": ["-10", "-2"], "C": ["-6", "-4"], "incircle_radius_exact": "sqrt(-sqrt(313)/2 + sqrt(5) + sqrt(377)/2)*sqrt(-sqrt(377)/2 + sqrt(5) + sqrt(313)/2)*sqrt(-sqrt(5) + sqrt(313)/2 + sqrt(377)/2)/sqrt(sqrt(5) + sqrt(313)/2 + sqrt(377)/2)", "incircle_radius_approx": 1.8277829342539866, "source_dataset": "advanced_geometry", "source_index": 29, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -5), B=(1, 7), and C=(9, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(18.000, 7.000)", "metadata": {"A": ["9", "-5"], "B": ["1", "7"], "C": ["9", "1"], "ortho": ["18", "7"], "orthocenter_approx": [18.0, 7.0], "source_dataset": "advanced_geometry", "source_index": 30, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 6), B=(5, 8), and C=(-5, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.087", "metadata": {"A": ["-7", "6"], "B": ["5", "8"], "C": ["-5", "4"], "incircle_radius_exact": "sqrt(-sqrt(2) + sqrt(29) + sqrt(37))*sqrt(-sqrt(29) + sqrt(2) + sqrt(37))*sqrt(-sqrt(37) + sqrt(2) + sqrt(29))/sqrt(sqrt(2) + sqrt(29) + sqrt(37))", "incircle_radius_approx": 1.0867758790164321, "source_dataset": "advanced_geometry", "source_index": 31, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -7), B=(0, -7), and C=(-1, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "116.57°", "metadata": {"A": ["6", "-7"], "B": ["0", "-7"], "C": ["-1", "-5"], "angle_ABC_degrees": 116.56505117707799, "source_dataset": "advanced_geometry", "source_index": 32, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 3), B=(4, -9), and C=(6, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.080", "metadata": {"A": ["8", "3"], "B": ["4", "-9"], "C": ["6", "5"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + 6*sqrt(2))*sqrt(-4*sqrt(2) + 2*sqrt(10))*sqrt(4*sqrt(2) + 2*sqrt(10))/sqrt(2*sqrt(10) + 6*sqrt(2))", "incircle_radius_approx": 1.080363026950906, "source_dataset": "advanced_geometry", "source_index": 33, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, 2), B=(-4, -1), and C=(-9, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.679", "metadata": {"A": ["4", "2"], "B": ["-4", "-1"], "C": ["-9", "3"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + sqrt(41)/2 + sqrt(73)/2)*sqrt(-sqrt(41)/2 + sqrt(73)/2 + sqrt(170)/2)*sqrt(-sqrt(73)/2 + sqrt(41)/2 + sqrt(170)/2)/sqrt(sqrt(41)/2 + sqrt(73)/2 + sqrt(170)/2)", "incircle_radius_approx": 1.6794391712097267, "source_dataset": "advanced_geometry", "source_index": 34, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -4), B=(-3, -4), and C=(-7, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.576", "metadata": {"A": ["0", "-4"], "B": ["-3", "-4"], "C": ["-7", "-1"], "incircle_radius_exact": "sqrt(-1 + sqrt(58)/2)*sqrt(1 + sqrt(58)/2)*sqrt(4 - sqrt(58)/2)/sqrt(sqrt(58)/2 + 4)", "incircle_radius_approx": 0.5763403412041376, "source_dataset": "advanced_geometry", "source_index": 35, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, -1), B=(-2, 3), and C=(-9, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.222, 2.778)", "metadata": {"A": ["-7", "-1"], "B": ["-2", "3"], "C": ["-9", "10"], "ortho": ["-29/9", "25/9"], "orthocenter_approx": [-3.2222222222222223, 2.7777777777777777], "source_dataset": "advanced_geometry", "source_index": 36, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 1), B=(9, -5), and C=(-3, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.355, 3.129)", "metadata": {"A": ["4", "1"], "B": ["9", "-5"], "C": ["-3", "-3"], "ortho": ["135/31", "97/31"], "orthocenter_approx": [4.354838709677419, 3.129032258064516], "source_dataset": "advanced_geometry", "source_index": 37, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 7), B=(3, 1), and C=(1, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.448", "metadata": {"A": ["2", "7"], "B": ["3", "1"], "C": ["1", "7"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(37)/2 + sqrt(10))*sqrt(-sqrt(10) + 1/2 + sqrt(37)/2)*sqrt(-sqrt(37)/2 + 1/2 + sqrt(10))/sqrt(1/2 + sqrt(37)/2 + sqrt(10))", "incircle_radius_approx": 0.44751680140974925, "source_dataset": "advanced_geometry", "source_index": 38, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, 0), B=(7, 7), and C=(2, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["4", "0"], "B": ["7", "7"], "C": ["2", "5"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 39, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, 6), B=(-1, -5), and C=(-5, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.161, -3.548)", "metadata": {"A": ["5", "6"], "B": ["-1", "-5"], "C": ["-5", "-2"], "ortho": ["-67/31", "-110/31"], "orthocenter_approx": [-2.161290322580645, -3.5483870967741935], "source_dataset": "advanced_geometry", "source_index": 40, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, -8), B=(-6, 2), and C=(-9, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.629", "metadata": {"A": ["4", "-8"], "B": ["-6", "2"], "C": ["-9", "-9"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + sqrt(170)/2 + 5*sqrt(2))*sqrt(-sqrt(170)/2 + sqrt(130)/2 + 5*sqrt(2))*sqrt(-5*sqrt(2) + sqrt(130)/2 + sqrt(170)/2)/sqrt(sqrt(130)/2 + sqrt(170)/2 + 5*sqrt(2))", "incircle_radius_approx": 3.628607399910975, "source_dataset": "advanced_geometry", "source_index": 41, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-1, -9), B=(5, -10), and C=(6, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "86.25°", "metadata": {"A": ["-1", "-9"], "B": ["5", "-10"], "C": ["6", "0"], "angle_ABC_degrees": 86.24827092947403, "source_dataset": "advanced_geometry", "source_index": 42, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -7), B=(0, -4), and C=(-3, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.905, -13.714)", "metadata": {"A": ["9", "-7"], "B": ["0", "-4"], "C": ["-3", "4"], "ortho": ["-187/21", "-96/7"], "orthocenter_approx": [-8.904761904761905, -13.714285714285714], "source_dataset": "advanced_geometry", "source_index": 43, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 0), B=(-10, -8), and C=(6, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(13.371, -5.663)", "metadata": {"A": ["7", "0"], "B": ["-10", "-8"], "C": ["6", "10"], "ortho": ["1190/89", "-504/89"], "orthocenter_approx": [13.370786516853933, -5.662921348314606], "source_dataset": "advanced_geometry", "source_index": 44, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 7), B=(-7, -2), and C=(0, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-21.333, 17.111)", "metadata": {"A": ["-4", "7"], "B": ["-7", "-2"], "C": ["0", "10"], "ortho": ["-64/3", "154/9"], "orthocenter_approx": [-21.333333333333332, 17.11111111111111], "source_dataset": "advanced_geometry", "source_index": 45, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, 0), B=(-5, 10), and C=(-10, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.165", "metadata": {"A": ["-4", "0"], "B": ["-5", "10"], "C": ["-10", "6"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(41)/2 + 3*sqrt(2))*sqrt(-3*sqrt(2) + sqrt(41)/2 + sqrt(101)/2)*sqrt(-sqrt(41)/2 + 3*sqrt(2) + sqrt(101)/2)/sqrt(sqrt(41)/2 + 3*sqrt(2) + sqrt(101)/2)", "incircle_radius_approx": 2.16534569868405, "source_dataset": "advanced_geometry", "source_index": 46, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -2), B=(-6, 1), and C=(-8, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.324, 2.378)", "metadata": {"A": ["9", "-2"], "B": ["-6", "1"], "C": ["-8", "-6"], "ortho": ["-234/37", "88/37"], "orthocenter_approx": [-6.324324324324325, 2.3783783783783785], "source_dataset": "advanced_geometry", "source_index": 47, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, 5), B=(8, 8), and C=(-1, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.695, 24.373)", "metadata": {"A": ["-8", "5"], "B": ["8", "8"], "C": ["-1", "10"], "ortho": ["-218/59", "1438/59"], "orthocenter_approx": [-3.694915254237288, 24.372881355932204], "source_dataset": "advanced_geometry", "source_index": 48, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, -5), B=(-5, 2), and C=(4, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.416, 6.629)", "metadata": {"A": ["8", "-5"], "B": ["-5", "2"], "C": ["4", "4"], "ortho": ["482/89", "590/89"], "orthocenter_approx": [5.415730337078652, 6.629213483146067], "source_dataset": "advanced_geometry", "source_index": 49, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, 8), B=(7, 7), and C=(8, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.387", "metadata": {"A": ["6", "8"], "B": ["7", "7"], "C": ["8", "-2"], "incircle_radius_exact": "sqrt(-sqrt(2)/2 + sqrt(82)/2 + sqrt(26))*sqrt(-sqrt(26) + sqrt(2)/2 + sqrt(82)/2)*sqrt(-sqrt(82)/2 + sqrt(2)/2 + sqrt(26))/sqrt(sqrt(2)/2 + sqrt(82)/2 + sqrt(26))", "incircle_radius_approx": 0.3870785866001241, "source_dataset": "advanced_geometry", "source_index": 50, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(8, 1), B=(-7, -8), and C=(1, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.178", "metadata": {"A": ["8", "1"], "B": ["-7", "-8"], "C": ["1", "-6"], "incircle_radius_exact": "sqrt(-sqrt(17) + 7*sqrt(2)/2 + 3*sqrt(34)/2)*sqrt(-7*sqrt(2)/2 + sqrt(17) + 3*sqrt(34)/2)*sqrt(-3*sqrt(34)/2 + sqrt(17) + 7*sqrt(2)/2)/sqrt(sqrt(17) + 7*sqrt(2)/2 + 3*sqrt(34)/2)", "incircle_radius_approx": 1.178498732647984, "source_dataset": "advanced_geometry", "source_index": 51, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, -7), B=(-2, -1), and C=(3, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "55.62°", "metadata": {"A": ["-5", "-7"], "B": ["-2", "-1"], "C": ["3", "-10"], "angle_ABC_degrees": 55.61965527615514, "source_dataset": "advanced_geometry", "source_index": 52, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, 7), B=(-6, 5), and C=(-10, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "119.45°", "metadata": {"A": ["6", "7"], "B": ["-6", "5"], "C": ["-10", "-6"], "angle_ABC_degrees": 119.44542872992561, "source_dataset": "advanced_geometry", "source_index": 53, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, 2), B=(-4, -7), and C=(0, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.738", "metadata": {"A": ["0", "2"], "B": ["-4", "-7"], "C": ["0", "-9"], "incircle_radius_exact": "sqrt(-11/2 + sqrt(5) + sqrt(97)/2)*sqrt(-sqrt(5) + sqrt(97)/2 + 11/2)*sqrt(-sqrt(97)/2 + sqrt(5) + 11/2)/sqrt(sqrt(5) + sqrt(97)/2 + 11/2)", "incircle_radius_approx": 1.7376885134372428, "source_dataset": "advanced_geometry", "source_index": 54, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, 1), B=(-9, 8), and C=(-10, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.421, 11.421)", "metadata": {"A": ["3", "1"], "B": ["-9", "8"], "C": ["-10", "7"], "ortho": ["-141/19", "217/19"], "orthocenter_approx": [-7.421052631578947, 11.421052631578947], "source_dataset": "advanced_geometry", "source_index": 55, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, -9), B=(1, 3), and C=(-9, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.299", "metadata": {"A": ["8", "-9"], "B": ["1", "3"], "C": ["-9", "5"], "incircle_radius_exact": "sqrt(-sqrt(193)/2 + sqrt(26) + sqrt(485)/2)*sqrt(-sqrt(26) + sqrt(193)/2 + sqrt(485)/2)*sqrt(-sqrt(485)/2 + sqrt(26) + sqrt(193)/2)/sqrt(sqrt(26) + sqrt(193)/2 + sqrt(485)/2)", "incircle_radius_approx": 2.298691118922622, "source_dataset": "advanced_geometry", "source_index": 56, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -4), B=(9, -5), and C=(-3, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.082", "metadata": {"A": ["2", "-4"], "B": ["9", "-5"], "C": ["-3", "-3"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(26)/2 + sqrt(37))*sqrt(-sqrt(26)/2 + 5*sqrt(2)/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(26)/2 + 5*sqrt(2)/2)/sqrt(sqrt(26)/2 + 5*sqrt(2)/2 + sqrt(37))", "incircle_radius_approx": 0.08218408348524164, "source_dataset": "advanced_geometry", "source_index": 57, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, 4), B=(-5, 2), and C=(1, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(13.667, -54.000)", "metadata": {"A": ["4", "4"], "B": ["-5", "2"], "C": ["1", "3"], "ortho": ["41/3", "-54"], "orthocenter_approx": [13.666666666666666, -54.0], "source_dataset": "advanced_geometry", "source_index": 58, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-4, 7), B=(1, 4), and C=(-9, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.65°", "metadata": {"A": ["-4", "7"], "B": ["1", "4"], "C": ["-9", "6"], "angle_ABC_degrees": 19.653824058053306, "source_dataset": "advanced_geometry", "source_index": 59, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 1), B=(-5, 7), and C=(9, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(14.143, 7.000)", "metadata": {"A": ["9", "1"], "B": ["-5", "7"], "C": ["9", "-5"], "ortho": ["99/7", "7"], "orthocenter_approx": [14.142857142857142, 7.0], "source_dataset": "advanced_geometry", "source_index": 60, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, -4), B=(-8, 9), and C=(-7, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "92.29°", "metadata": {"A": ["4", "-4"], "B": ["-8", "9"], "C": ["-7", "10"], "angle_ABC_degrees": 92.29061004263853, "source_dataset": "advanced_geometry", "source_index": 61, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, -5), B=(-7, -6), and C=(-7, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.000, -5.000)", "metadata": {"A": ["-10", "-5"], "B": ["-7", "-6"], "C": ["-7", "-8"], "ortho": ["-6", "-5"], "orthocenter_approx": [-6.0, -5.0], "source_dataset": "advanced_geometry", "source_index": 62, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 8), B=(-7, -10), and C=(-1, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.41°", "metadata": {"A": ["1", "8"], "B": ["-7", "-10"], "C": ["-1", "6"], "angle_ABC_degrees": 3.406443754994719, "source_dataset": "advanced_geometry", "source_index": 63, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, -10), B=(7, 6), and C=(5, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.47°", "metadata": {"A": ["5", "-10"], "B": ["7", "6"], "C": ["5", "-9"], "angle_ABC_degrees": 0.46962701968964715, "source_dataset": "advanced_geometry", "source_index": 64, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 8), B=(3, 5), and C=(-1, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-11.778, 15.556)", "metadata": {"A": ["9", "8"], "B": ["3", "5"], "C": ["-1", "-6"], "ortho": ["-106/9", "140/9"], "orthocenter_approx": [-11.777777777777779, 15.555555555555555], "source_dataset": "advanced_geometry", "source_index": 65, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, 2), B=(-2, 2), and C=(-4, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.000, 1.200)", "metadata": {"A": ["-8", "2"], "B": ["-2", "2"], "C": ["-4", "-8"], "ortho": ["-4", "6/5"], "orthocenter_approx": [-4.0, 1.2], "source_dataset": "advanced_geometry", "source_index": 66, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 4), B=(4, 7), and C=(-9, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.503", "metadata": {"A": ["-3", "4"], "B": ["4", "7"], "C": ["-9", "-7"], "incircle_radius_exact": "sqrt(-sqrt(157)/2 + sqrt(58)/2 + sqrt(365)/2)*sqrt(-sqrt(365)/2 + sqrt(58)/2 + sqrt(157)/2)*sqrt(-sqrt(58)/2 + sqrt(157)/2 + sqrt(365)/2)/sqrt(sqrt(58)/2 + sqrt(157)/2 + sqrt(365)/2)", "incircle_radius_approx": 1.5031575084634152, "source_dataset": "advanced_geometry", "source_index": 67, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, -3), B=(3, 5), and C=(-2, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.293", "metadata": {"A": ["9", "-3"], "B": ["3", "5"], "C": ["-2", "10"], "incircle_radius_exact": "sqrt(-5 + 5*sqrt(2)/2 + sqrt(290)/2)*sqrt(-5*sqrt(2)/2 + 5 + sqrt(290)/2)*sqrt(-sqrt(290)/2 + 5*sqrt(2)/2 + 5)/sqrt(5*sqrt(2)/2 + 5 + sqrt(290)/2)", "incircle_radius_approx": 0.2932512261526581, "source_dataset": "advanced_geometry", "source_index": 68, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 5), B=(-4, 2), and C=(3, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(24.000, -19.000)", "metadata": {"A": ["0", "5"], "B": ["-4", "2"], "C": ["3", "9"], "ortho": ["24", "-19"], "orthocenter_approx": [24.0, -19.0], "source_dataset": "advanced_geometry", "source_index": 69, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 9), B=(-1, 4), and C=(3, -5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.885", "metadata": {"A": ["8", "9"], "B": ["-1", "4"], "C": ["3", "-5"], "incircle_radius_exact": "sqrt(-sqrt(106)/2 + sqrt(97)/2 + sqrt(221)/2)*sqrt(-sqrt(221)/2 + sqrt(97)/2 + sqrt(106)/2)*sqrt(-sqrt(97)/2 + sqrt(106)/2 + sqrt(221)/2)/sqrt(sqrt(97)/2 + sqrt(106)/2 + sqrt(221)/2)", "incircle_radius_approx": 2.8848441598346626, "source_dataset": "advanced_geometry", "source_index": 70, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -2), B=(-10, 5), and C=(7, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(18.308, 33.308)", "metadata": {"A": ["10", "-2"], "B": ["-10", "5"], "C": ["7", "1"], "ortho": ["238/13", "433/13"], "orthocenter_approx": [18.307692307692307, 33.30769230769231], "source_dataset": "advanced_geometry", "source_index": 71, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -7), B=(0, -3), and C=(10, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.765, -3.471)", "metadata": {"A": ["8", "-7"], "B": ["0", "-3"], "C": ["10", "9"], "ortho": ["64/17", "-59/17"], "orthocenter_approx": [3.764705882352941, -3.4705882352941178], "source_dataset": "advanced_geometry", "source_index": 72, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -7), B=(7, -7), and C=(0, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.737", "metadata": {"A": ["-4", "-7"], "B": ["7", "-7"], "C": ["0", "7"], "incircle_radius_exact": "sqrt(-11/2 + sqrt(53) + 7*sqrt(5)/2)*sqrt(-7*sqrt(5)/2 + 11/2 + sqrt(53))*sqrt(-sqrt(53) + 11/2 + 7*sqrt(5)/2)/sqrt(11/2 + sqrt(53) + 7*sqrt(5)/2)", "incircle_radius_approx": 3.7367126240901958, "source_dataset": "advanced_geometry", "source_index": 73, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, 7), B=(-1, -10), and C=(10, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "44.28°", "metadata": {"A": ["9", "7"], "B": ["-1", "-10"], "C": ["10", "-7"], "angle_ABC_degrees": 44.27933637748235, "source_dataset": "advanced_geometry", "source_index": 74, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-5, -5), B=(-4, 2), and C=(-5, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(30.000, 2.000)", "metadata": {"A": ["-5", "-5"], "B": ["-4", "2"], "C": ["-5", "7"], "ortho": ["30", "2"], "orthocenter_approx": [30.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 75, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, -5), B=(-1, -7), and C=(1, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "61.26°", "metadata": {"A": ["10", "-5"], "B": ["-1", "-7"], "C": ["1", "-1"], "angle_ABC_degrees": 61.26020470831195, "source_dataset": "advanced_geometry", "source_index": 76, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 7), B=(7, 3), and C=(-7, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.124", "metadata": {"A": ["-5", "7"], "B": ["7", "3"], "C": ["-7", "-8"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + sqrt(229)/2 + sqrt(317)/2)*sqrt(-sqrt(229)/2 + 2*sqrt(10) + sqrt(317)/2)*sqrt(-sqrt(317)/2 + 2*sqrt(10) + sqrt(229)/2)/sqrt(2*sqrt(10) + sqrt(229)/2 + sqrt(317)/2)", "incircle_radius_approx": 4.124041480096933, "source_dataset": "advanced_geometry", "source_index": 77, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -4), B=(10, 8), and C=(4, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.73°", "metadata": {"A": ["-3", "-4"], "B": ["10", "8"], "C": ["4", "-8"], "angle_ABC_degrees": 26.734564823055067, "source_dataset": "advanced_geometry", "source_index": 78, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 8), B=(2, 10), and C=(-5, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.500, 4.500)", "metadata": {"A": ["-4", "8"], "B": ["2", "10"], "C": ["-5", "9"], "ortho": ["-7/2", "9/2"], "orthocenter_approx": [-3.5, 4.5], "source_dataset": "advanced_geometry", "source_index": 79, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 9), B=(-1, -8), and C=(1, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.81°", "metadata": {"A": ["-4", "9"], "B": ["-1", "-8"], "C": ["1", "-3"], "angle_ABC_degrees": 31.80938928779315, "source_dataset": "advanced_geometry", "source_index": 80, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -8), B=(3, -2), and C=(-7, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.680, -20.400)", "metadata": {"A": ["-2", "-8"], "B": ["3", "-2"], "C": ["-7", "-9"], "ortho": ["167/25", "-102/5"], "orthocenter_approx": [6.68, -20.4], "source_dataset": "advanced_geometry", "source_index": 81, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -10), B=(-5, -9), and C=(3, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.910", "metadata": {"A": ["7", "-10"], "B": ["-5", "-9"], "C": ["3", "-2"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + 2*sqrt(5) + sqrt(145)/2)*sqrt(-sqrt(145)/2 + 2*sqrt(5) + sqrt(113)/2)*sqrt(-2*sqrt(5) + sqrt(113)/2 + sqrt(145)/2)/sqrt(2*sqrt(5) + sqrt(113)/2 + sqrt(145)/2)", "incircle_radius_approx": 2.9099178961149113, "source_dataset": "advanced_geometry", "source_index": 82, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, 9), B=(-1, 8), and C=(-5, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.444, 8.111)", "metadata": {"A": ["-3", "9"], "B": ["-1", "8"], "C": ["-5", "1"], "ortho": ["-13/9", "73/9"], "orthocenter_approx": [-1.4444444444444444, 8.11111111111111], "source_dataset": "advanced_geometry", "source_index": 83, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, 8), B=(9, 4), and C=(-3, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "14.04°", "metadata": {"A": ["-9", "8"], "B": ["9", "4"], "C": ["-3", "10"], "angle_ABC_degrees": 14.036243467926477, "source_dataset": "advanced_geometry", "source_index": 84, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 8), B=(8, -7), and C=(-10, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(12.247, -11.483)", "metadata": {"A": ["9", "8"], "B": ["8", "-7"], "C": ["-10", "-10"], "ortho": ["1090/89", "-1022/89"], "orthocenter_approx": [12.247191011235955, -11.48314606741573], "source_dataset": "advanced_geometry", "source_index": 85, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, 6), B=(1, -6), and C=(5, -10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-12.538, -8.538)", "metadata": {"A": ["2", "6"], "B": ["1", "-6"], "C": ["5", "-10"], "ortho": ["-163/13", "-111/13"], "orthocenter_approx": [-12.538461538461538, -8.538461538461538], "source_dataset": "advanced_geometry", "source_index": 86, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, -2), B=(-9, 2), and C=(10, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.19°", "metadata": {"A": ["6", "-2"], "B": ["-9", "2"], "C": ["10", "-3"], "angle_ABC_degrees": 0.1878543416668164, "source_dataset": "advanced_geometry", "source_index": 87, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, -2), B=(4, 10), and C=(4, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.76°", "metadata": {"A": ["3", "-2"], "B": ["4", "10"], "C": ["4", "-10"], "angle_ABC_degrees": 4.763641690726178, "source_dataset": "advanced_geometry", "source_index": 88, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, 6), B=(-8, 2), and C=(-1, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.411", "metadata": {"A": ["1", "6"], "B": ["-8", "2"], "C": ["-1", "-10"], "incircle_radius_exact": "sqrt(-sqrt(193)/2 + sqrt(97)/2 + sqrt(65))*sqrt(-sqrt(65) + sqrt(97)/2 + sqrt(193)/2)*sqrt(-sqrt(97)/2 + sqrt(193)/2 + sqrt(65))/sqrt(sqrt(97)/2 + sqrt(193)/2 + sqrt(65))", "incircle_radius_approx": 3.4114439199387214, "source_dataset": "advanced_geometry", "source_index": 89, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, 3), B=(-9, -5), and C=(-9, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.684", "metadata": {"A": ["-6", "3"], "B": ["-9", "-5"], "C": ["-9", "-1"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(73)/2)*sqrt(1/2 + sqrt(73)/2)*sqrt(9/2 - sqrt(73)/2)/sqrt(sqrt(73)/2 + 9/2)", "incircle_radius_approx": 0.6839943820237032, "source_dataset": "advanced_geometry", "source_index": 90, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 5), B=(0, 9), and C=(0, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.239", "metadata": {"A": ["1", "5"], "B": ["0", "9"], "C": ["0", "7"], "incircle_radius_exact": "sqrt(-1 + sqrt(5)/2 + sqrt(17)/2)*sqrt(-sqrt(17)/2 + 1 + sqrt(5)/2)*sqrt(-sqrt(5)/2 + 1 + sqrt(17)/2)/sqrt(1 + sqrt(5)/2 + sqrt(17)/2)", "incircle_radius_approx": 0.23925810073547518, "source_dataset": "advanced_geometry", "source_index": 91, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -4), B=(7, 10), and C=(-6, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.158", "metadata": {"A": ["6", "-4"], "B": ["7", "10"], "C": ["-6", "-10"], "incircle_radius_exact": "sqrt(-sqrt(197)/2 + 3*sqrt(5) + sqrt(569)/2)*sqrt(-3*sqrt(5) + sqrt(197)/2 + sqrt(569)/2)*sqrt(-sqrt(569)/2 + 3*sqrt(5) + sqrt(197)/2)/sqrt(3*sqrt(5) + sqrt(197)/2 + sqrt(569)/2)", "incircle_radius_approx": 3.1575378921983988, "source_dataset": "advanced_geometry", "source_index": 92, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 10), B=(10, 8), and C=(-2, 5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.57°", "metadata": {"A": ["1", "10"], "B": ["10", "8"], "C": ["-2", "5"], "angle_ABC_degrees": 26.56505117707799, "source_dataset": "advanced_geometry", "source_index": 93, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, 6), B=(-9, 3), and C=(5, -4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "40.60°", "metadata": {"A": ["3", "6"], "B": ["-9", "3"], "C": ["5", "-4"], "angle_ABC_degrees": 40.60129464500447, "source_dataset": "advanced_geometry", "source_index": 94, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-5, 3), B=(0, -4), and C=(-3, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.500, 2.500)", "metadata": {"A": ["-5", "3"], "B": ["0", "-4"], "C": ["-3", "5"], "ortho": ["-13/2", "5/2"], "orthocenter_approx": [-6.5, 2.5], "source_dataset": "advanced_geometry", "source_index": 95, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, 4), B=(-1, -4), and C=(6, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.440, -1.040)", "metadata": {"A": ["2", "4"], "B": ["-1", "-4"], "C": ["6", "-2"], "ortho": ["86/25", "-26/25"], "orthocenter_approx": [3.44, -1.04], "source_dataset": "advanced_geometry", "source_index": 96, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 7), B=(9, -1), and C=(7, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(85.500, -10.000)", "metadata": {"A": ["9", "7"], "B": ["9", "-1"], "C": ["7", "-10"], "ortho": ["171/2", "-10"], "orthocenter_approx": [85.5, -10.0], "source_dataset": "advanced_geometry", "source_index": 97, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, -2), B=(-10, 7), and C=(-9, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-20.000, -3.000)", "metadata": {"A": ["-10", "-2"], "B": ["-10", "7"], "C": ["-9", "-3"], "ortho": ["-20", "-3"], "orthocenter_approx": [-20.0, -3.0], "source_dataset": "advanced_geometry", "source_index": 98, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, 8), B=(5, -5), and C=(8, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "67.52°", "metadata": {"A": ["-10", "8"], "B": ["5", "-5"], "C": ["8", "4"], "angle_ABC_degrees": 67.5205656028969, "source_dataset": "advanced_geometry", "source_index": 99, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, 5), B=(-5, 5), and C=(-1, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["10", "5"], "B": ["-5", "5"], "C": ["-1", "1"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 100, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, -10), B=(-3, -6), and C=(-3, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.667, -10.000)", "metadata": {"A": ["-9", "-10"], "B": ["-3", "-6"], "C": ["-3", "-8"], "ortho": ["-5/3", "-10"], "orthocenter_approx": [-1.6666666666666667, -10.0], "source_dataset": "advanced_geometry", "source_index": 101, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 4), B=(-8, -5), and C=(2, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.432", "metadata": {"A": ["-9", "4"], "B": ["-8", "-5"], "C": ["2", "-8"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + sqrt(82)/2 + sqrt(265)/2)*sqrt(-sqrt(265)/2 + sqrt(82)/2 + sqrt(109)/2)*sqrt(-sqrt(82)/2 + sqrt(109)/2 + sqrt(265)/2)/sqrt(sqrt(82)/2 + sqrt(109)/2 + sqrt(265)/2)", "incircle_radius_approx": 2.431898984637147, "source_dataset": "advanced_geometry", "source_index": 102, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -10), B=(4, -10), and C=(0, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.924", "metadata": {"A": ["6", "-10"], "B": ["4", "-10"], "C": ["0", "10"], "incircle_radius_exact": "sqrt(-1 + 2*sqrt(26) + sqrt(109))*sqrt(-sqrt(109) + 1 + 2*sqrt(26))*sqrt(-2*sqrt(26) + 1 + sqrt(109))/sqrt(1 + 2*sqrt(26) + sqrt(109))", "incircle_radius_approx": 0.9242850830086291, "source_dataset": "advanced_geometry", "source_index": 103, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, 9), B=(0, -8), and C=(-5, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(27.455, 19.455)", "metadata": {"A": ["-6", "9"], "B": ["0", "-8"], "C": ["-5", "8"], "ortho": ["302/11", "214/11"], "orthocenter_approx": [27.454545454545453, 19.454545454545453], "source_dataset": "advanced_geometry", "source_index": 104, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, -6), B=(2, 4), and C=(9, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "51.34°", "metadata": {"A": ["10", "-6"], "B": ["2", "4"], "C": ["9", "4"], "angle_ABC_degrees": 51.340191745909905, "source_dataset": "advanced_geometry", "source_index": 105, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, -9), B=(2, 1), and C=(-1, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.795", "metadata": {"A": ["10", "-9"], "B": ["2", "1"], "C": ["-1", "-2"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + 3*sqrt(2)/2 + sqrt(41))*sqrt(-3*sqrt(2)/2 + sqrt(41) + sqrt(170)/2)*sqrt(-sqrt(41) + 3*sqrt(2)/2 + sqrt(170)/2)/sqrt(3*sqrt(2)/2 + sqrt(41) + sqrt(170)/2)", "incircle_radius_approx": 1.7947775579137595, "source_dataset": "advanced_geometry", "source_index": 106, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, -2), B=(2, -6), and C=(-6, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.119, -7.017)", "metadata": {"A": ["-7", "-2"], "B": ["2", "-6"], "C": ["-6", "-9"], "ortho": ["-302/59", "-414/59"], "orthocenter_approx": [-5.11864406779661, -7.016949152542373], "source_dataset": "advanced_geometry", "source_index": 107, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -4), B=(-3, -1), and C=(5, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.053", "metadata": {"A": ["2", "-4"], "B": ["-3", "-1"], "C": ["5", "-6"], "incircle_radius_exact": "sqrt(-sqrt(13)/2 + sqrt(34)/2 + sqrt(89)/2)*sqrt(-sqrt(34)/2 + sqrt(13)/2 + sqrt(89)/2)*sqrt(-sqrt(89)/2 + sqrt(13)/2 + sqrt(34)/2)/sqrt(sqrt(13)/2 + sqrt(34)/2 + sqrt(89)/2)", "incircle_radius_approx": 0.052992810570029976, "source_dataset": "advanced_geometry", "source_index": 108, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(3, 10), B=(-4, 1), and C=(-1, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "52.13°", "metadata": {"A": ["3", "10"], "B": ["-4", "1"], "C": ["-1", "1"], "angle_ABC_degrees": 52.125016348901795, "source_dataset": "advanced_geometry", "source_index": 109, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, -6), B=(2, -4), and C=(-8, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.747", "metadata": {"A": ["10", "-6"], "B": ["2", "-4"], "C": ["-8", "9"], "incircle_radius_exact": "sqrt(-sqrt(17) + sqrt(269)/2 + 3*sqrt(61)/2)*sqrt(-sqrt(269)/2 + sqrt(17) + 3*sqrt(61)/2)*sqrt(-3*sqrt(61)/2 + sqrt(17) + sqrt(269)/2)/sqrt(sqrt(17) + sqrt(269)/2 + 3*sqrt(61)/2)", "incircle_radius_approx": 1.7471543316345512, "source_dataset": "advanced_geometry", "source_index": 110, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, -9), B=(-2, 3), and C=(8, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.877", "metadata": {"A": ["-8", "-9"], "B": ["-2", "3"], "C": ["8", "8"], "incircle_radius_exact": "sqrt(-sqrt(5)/2 + sqrt(545)/2)*sqrt(sqrt(5)/2 + sqrt(545)/2)*sqrt(-sqrt(545)/2 + 11*sqrt(5)/2)/sqrt(sqrt(545)/2 + 11*sqrt(5)/2)", "incircle_radius_approx": 1.877269038960274, "source_dataset": "advanced_geometry", "source_index": 111, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, 5), B=(-4, 4), and C=(6, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.340, 5.957)", "metadata": {"A": ["2", "5"], "B": ["-4", "4"], "C": ["6", "-10"], "ortho": ["157/47", "280/47"], "orthocenter_approx": [3.3404255319148937, 5.957446808510638], "source_dataset": "advanced_geometry", "source_index": 112, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 7), B=(-1, 3), and C=(7, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.18°", "metadata": {"A": ["6", "7"], "B": ["-1", "3"], "C": ["7", "7"], "angle_ABC_degrees": 3.1798301198642345, "source_dataset": "advanced_geometry", "source_index": 113, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -3), B=(1, -6), and C=(-3, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.996", "metadata": {"A": ["3", "-3"], "B": ["1", "-6"], "C": ["-3", "-5"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(13)/2 + sqrt(17)/2)*sqrt(-sqrt(13)/2 + sqrt(17)/2 + sqrt(10))*sqrt(-sqrt(17)/2 + sqrt(13)/2 + sqrt(10))/sqrt(sqrt(13)/2 + sqrt(17)/2 + sqrt(10))", "incircle_radius_approx": 0.9962135189748783, "source_dataset": "advanced_geometry", "source_index": 114, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, -3), B=(-5, -10), and C=(-10, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "79.45°", "metadata": {"A": ["5", "-3"], "B": ["-5", "-10"], "C": ["-10", "1"], "angle_ABC_degrees": 79.45193458185787, "source_dataset": "advanced_geometry", "source_index": 115, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -7), B=(-9, 3), and C=(8, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "49.03°", "metadata": {"A": ["-3", "-7"], "B": ["-9", "3"], "C": ["8", "0"], "angle_ABC_degrees": 49.028263666485145, "source_dataset": "advanced_geometry", "source_index": 116, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 4), B=(-3, 6), and C=(-4, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.721", "metadata": {"A": ["-5", "4"], "B": ["-3", "6"], "C": ["-4", "9"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(2) + sqrt(26)/2)*sqrt(-sqrt(2) + sqrt(10)/2 + sqrt(26)/2)*sqrt(-sqrt(26)/2 + sqrt(2) + sqrt(10)/2)/sqrt(sqrt(2) + sqrt(10)/2 + sqrt(26)/2)", "incircle_radius_approx": 0.7213885381331597, "source_dataset": "advanced_geometry", "source_index": 117, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 0), B=(2, -5), and C=(8, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.000, 13.000)", "metadata": {"A": ["6", "0"], "B": ["2", "-5"], "C": ["8", "1"], "ortho": ["-7", "13"], "orthocenter_approx": [-7.0, 13.0], "source_dataset": "advanced_geometry", "source_index": 118, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 7), B=(-5, -3), and C=(10, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.04°", "metadata": {"A": ["-4", "7"], "B": ["-5", "-3"], "C": ["10", "8"], "angle_ABC_degrees": 48.035569125055574, "source_dataset": "advanced_geometry", "source_index": 119, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, -4), B=(6, -4), and C=(3, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.000, -6.333)", "metadata": {"A": ["10", "-4"], "B": ["6", "-4"], "C": ["3", "5"], "ortho": ["3", "-19/3"], "orthocenter_approx": [3.0, -6.333333333333333], "source_dataset": "advanced_geometry", "source_index": 120, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, 4), B=(8, -5), and C=(5, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "43.21°", "metadata": {"A": ["9", "4"], "B": ["8", "-5"], "C": ["5", "-1"], "angle_ABC_degrees": 43.210089391753925, "source_dataset": "advanced_geometry", "source_index": 121, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -2), B=(4, 5), and C=(9, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.680, 5.800)", "metadata": {"A": ["-1", "-2"], "B": ["4", "5"], "C": ["9", "2"], "ortho": ["92/25", "29/5"], "orthocenter_approx": [3.68, 5.8], "source_dataset": "advanced_geometry", "source_index": 122, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, 3), B=(-8, 5), and C=(-3, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-14.000, 0.500)", "metadata": {"A": ["-9", "3"], "B": ["-8", "5"], "C": ["-3", "-5"], "ortho": ["-14", "1/2"], "orthocenter_approx": [-14.0, 0.5], "source_dataset": "advanced_geometry", "source_index": 123, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, 9), B=(3, 3), and C=(-10, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(20.211, -35.246)", "metadata": {"A": ["10", "9"], "B": ["3", "3"], "C": ["-10", "0"], "ortho": ["384/19", "-2009/57"], "orthocenter_approx": [20.210526315789473, -35.24561403508772], "source_dataset": "advanced_geometry", "source_index": 124, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 0), B=(3, 0), and C=(-8, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.000, 0.000)", "metadata": {"A": ["-8", "0"], "B": ["3", "0"], "C": ["-8", "6"], "ortho": ["-8", "0"], "orthocenter_approx": [-8.0, 0.0], "source_dataset": "advanced_geometry", "source_index": 125, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, 6), B=(1, 2), and C=(-5, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "66.37°", "metadata": {"A": ["-7", "6"], "B": ["1", "2"], "C": ["-5", "-3"], "angle_ABC_degrees": 66.37062226934319, "source_dataset": "advanced_geometry", "source_index": 126, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 9), B=(-5, 7), and C=(6, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.912", "metadata": {"A": ["6", "9"], "B": ["-5", "7"], "C": ["6", "-7"], "incircle_radius_exact": "sqrt(-8 + 5*sqrt(5)/2 + sqrt(317)/2)*sqrt(-sqrt(317)/2 + 5*sqrt(5)/2 + 8)*sqrt(-5*sqrt(5)/2 + 8 + sqrt(317)/2)/sqrt(5*sqrt(5)/2 + 8 + sqrt(317)/2)", "incircle_radius_approx": 3.9124297127532346, "source_dataset": "advanced_geometry", "source_index": 127, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 0), B=(0, -8), and C=(-3, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.852", "metadata": {"A": ["8", "0"], "B": ["0", "-8"], "C": ["-3", "0"], "incircle_radius_exact": "sqrt(-11/2 + sqrt(73)/2 + 4*sqrt(2))*sqrt(-4*sqrt(2) + sqrt(73)/2 + 11/2)*sqrt(-sqrt(73)/2 + 11/2 + 4*sqrt(2))/sqrt(sqrt(73)/2 + 11/2 + 4*sqrt(2))", "incircle_radius_approx": 2.8517992294211223, "source_dataset": "advanced_geometry", "source_index": 128, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, 3), B=(-7, 8), and C=(4, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "20.71°", "metadata": {"A": ["2", "3"], "B": ["-7", "8"], "C": ["4", "-5"], "angle_ABC_degrees": 20.709037591649032, "source_dataset": "advanced_geometry", "source_index": 129, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -4), B=(-1, 1), and C=(-9, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.981", "metadata": {"A": ["-4", "-4"], "B": ["-1", "1"], "C": ["-9", "2"], "incircle_radius_exact": "sqrt(-sqrt(34)/2 + sqrt(61)/2 + sqrt(65)/2)*sqrt(-sqrt(61)/2 + sqrt(34)/2 + sqrt(65)/2)*sqrt(-sqrt(65)/2 + sqrt(34)/2 + sqrt(61)/2)/sqrt(sqrt(34)/2 + sqrt(61)/2 + sqrt(65)/2)", "incircle_radius_approx": 1.9812509779146668, "source_dataset": "advanced_geometry", "source_index": 130, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 6), B=(9, 1), and C=(-3, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.918", "metadata": {"A": ["5", "6"], "B": ["9", "1"], "C": ["-3", "3"], "incircle_radius_exact": "sqrt(-sqrt(37) + sqrt(41)/2 + sqrt(73)/2)*sqrt(-sqrt(41)/2 + sqrt(73)/2 + sqrt(37))*sqrt(-sqrt(73)/2 + sqrt(41)/2 + sqrt(37))/sqrt(sqrt(41)/2 + sqrt(73)/2 + sqrt(37))", "incircle_radius_approx": 1.91792370583816, "source_dataset": "advanced_geometry", "source_index": 131, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, -10), B=(5, 1), and C=(-8, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.129, -3.921)", "metadata": {"A": ["4", "-10"], "B": ["5", "1"], "C": ["-8", "-3"], "ortho": ["296/139", "-545/139"], "orthocenter_approx": [2.129496402877698, -3.920863309352518], "source_dataset": "advanced_geometry", "source_index": 132, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, -9), B=(8, 6), and C=(-4, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.295", "metadata": {"A": ["-3", "-9"], "B": ["8", "6"], "C": ["-4", "-2"], "incircle_radius_exact": "sqrt(-2*sqrt(13) + 5*sqrt(2)/2 + sqrt(346)/2)*sqrt(-5*sqrt(2)/2 + 2*sqrt(13) + sqrt(346)/2)*sqrt(-sqrt(346)/2 + 5*sqrt(2)/2 + 2*sqrt(13))/sqrt(5*sqrt(2)/2 + 2*sqrt(13) + sqrt(346)/2)", "incircle_radius_approx": 2.2945877471922578, "source_dataset": "advanced_geometry", "source_index": 133, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, -3), B=(-7, 2), and C=(10, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.809, 7.124)", "metadata": {"A": ["-9", "-3"], "B": ["-7", "2"], "C": ["10", "0"], "ortho": ["-695/89", "634/89"], "orthocenter_approx": [-7.808988764044944, 7.123595505617978], "source_dataset": "advanced_geometry", "source_index": 134, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 3), B=(8, -4), and C=(2, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(11.579, -4.895)", "metadata": {"A": ["5", "3"], "B": ["8", "-4"], "C": ["2", "-9"], "ortho": ["220/19", "-93/19"], "orthocenter_approx": [11.578947368421053, -4.894736842105263], "source_dataset": "advanced_geometry", "source_index": 135, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 4), B=(10, -3), and C=(0, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(11.870, 10.087)", "metadata": {"A": ["7", "4"], "B": ["10", "-3"], "C": ["0", "5"], "ortho": ["273/23", "232/23"], "orthocenter_approx": [11.869565217391305, 10.08695652173913], "source_dataset": "advanced_geometry", "source_index": 136, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, -8), B=(-9, 6), and C=(-1, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.111, 3.556)", "metadata": {"A": ["-7", "-8"], "B": ["-9", "6"], "C": ["-1", "4"], "ortho": ["-37/9", "32/9"], "orthocenter_approx": [-4.111111111111111, 3.5555555555555554], "source_dataset": "advanced_geometry", "source_index": 137, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, -10), B=(9, -2), and C=(-7, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.597, -1.356)", "metadata": {"A": ["-8", "-10"], "B": ["9", "-2"], "C": ["-7", "8"], "ortho": ["-387/149", "-202/149"], "orthocenter_approx": [-2.597315436241611, -1.3557046979865772], "source_dataset": "advanced_geometry", "source_index": 138, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -9), B=(0, 0), and C=(-4, -1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "38.09°", "metadata": {"A": ["-7", "-9"], "B": ["0", "0"], "C": ["-4", "-1"], "angle_ABC_degrees": 38.08877288097532, "source_dataset": "advanced_geometry", "source_index": 139, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, -3), B=(-3, 0), and C=(1, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.719", "metadata": {"A": ["1", "-3"], "B": ["-3", "0"], "C": ["1", "-1"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(17)/2)*sqrt(3/2 + sqrt(17)/2)*sqrt(7/2 - sqrt(17)/2)/sqrt(sqrt(17)/2 + 7/2)", "incircle_radius_approx": 0.7192235935955849, "source_dataset": "advanced_geometry", "source_index": 140, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, -10), B=(1, -1), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(8.333, -19.333)", "metadata": {"A": ["-8", "-10"], "B": ["1", "-1"], "C": ["-3", "-8"], "ortho": ["25/3", "-58/3"], "orthocenter_approx": [8.333333333333334, -19.333333333333332], "source_dataset": "advanced_geometry", "source_index": 141, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -3), B=(0, 0), and C=(0, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.567", "metadata": {"A": ["-10", "-3"], "B": ["0", "0"], "C": ["0", "9"], "incircle_radius_exact": "sqrt(-9/2 + sqrt(109)/2 + sqrt(61))*sqrt(-sqrt(109)/2 + 9/2 + sqrt(61))*sqrt(-sqrt(61) + 9/2 + sqrt(109)/2)/sqrt(9/2 + sqrt(109)/2 + sqrt(61))", "incircle_radius_approx": 2.566968949815858, "source_dataset": "advanced_geometry", "source_index": 142, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -6), B=(-1, 8), and C=(5, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.333, 6.667)", "metadata": {"A": ["-6", "-6"], "B": ["-1", "8"], "C": ["5", "5"], "ortho": ["1/3", "20/3"], "orthocenter_approx": [0.3333333333333333, 6.666666666666667], "source_dataset": "advanced_geometry", "source_index": 143, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, 7), B=(4, -7), and C=(5, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.480", "metadata": {"A": ["3", "7"], "B": ["4", "-7"], "C": ["5", "-7"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(197)/2 + 5*sqrt(2))*sqrt(-sqrt(197)/2 + 1/2 + 5*sqrt(2))*sqrt(-5*sqrt(2) + 1/2 + sqrt(197)/2)/sqrt(1/2 + sqrt(197)/2 + 5*sqrt(2))", "incircle_radius_approx": 0.4798167735254775, "source_dataset": "advanced_geometry", "source_index": 144, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -3), B=(5, 4), and C=(-5, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.806", "metadata": {"A": ["7", "-3"], "B": ["5", "4"], "C": ["-5", "8"], "incircle_radius_exact": "sqrt(-sqrt(265)/2 + sqrt(53)/2 + sqrt(29))*sqrt(-sqrt(29) + sqrt(53)/2 + sqrt(265)/2)*sqrt(-sqrt(53)/2 + sqrt(29) + sqrt(265)/2)/sqrt(sqrt(53)/2 + sqrt(29) + sqrt(265)/2)", "incircle_radius_approx": 1.8060395074064965, "source_dataset": "advanced_geometry", "source_index": 145, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -2), B=(-10, -3), and C=(-8, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.43°", "metadata": {"A": ["-3", "-2"], "B": ["-10", "-3"], "C": ["-8", "-2"], "angle_ABC_degrees": 18.43494882292201, "source_dataset": "advanced_geometry", "source_index": 146, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, -6), B=(8, 7), and C=(-7, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.482", "metadata": {"A": ["-2", "-6"], "B": ["8", "7"], "C": ["-7", "2"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(89)/2 + sqrt(269)/2)*sqrt(-sqrt(269)/2 + sqrt(89)/2 + 5*sqrt(10)/2)*sqrt(-sqrt(89)/2 + 5*sqrt(10)/2 + sqrt(269)/2)/sqrt(sqrt(89)/2 + 5*sqrt(10)/2 + sqrt(269)/2)", "incircle_radius_approx": 3.4816777035214095, "source_dataset": "advanced_geometry", "source_index": 147, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-5, 0), B=(1, -4), and C=(-3, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.895, -0.842)", "metadata": {"A": ["-5", "0"], "B": ["1", "-4"], "C": ["-3", "5"], "ortho": ["-131/19", "-16/19"], "orthocenter_approx": [-6.894736842105263, -0.8421052631578947], "source_dataset": "advanced_geometry", "source_index": 148, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, 6), B=(6, -6), and C=(2, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.80°", "metadata": {"A": ["7", "6"], "B": ["6", "-6"], "C": ["2", "10"], "angle_ABC_degrees": 18.799885158652653, "source_dataset": "advanced_geometry", "source_index": 149, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -8), B=(4, -1), and C=(3, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.633", "metadata": {"A": ["-1", "-8"], "B": ["4", "-1"], "C": ["3", "0"], "incircle_radius_exact": "sqrt(-sqrt(2)/2 + sqrt(74)/2 + 2*sqrt(5))*sqrt(-2*sqrt(5) + sqrt(2)/2 + sqrt(74)/2)*sqrt(-sqrt(74)/2 + sqrt(2)/2 + 2*sqrt(5))/sqrt(sqrt(2)/2 + sqrt(74)/2 + 2*sqrt(5))", "incircle_radius_approx": 0.6328843299434944, "source_dataset": "advanced_geometry", "source_index": 150, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, -5), B=(-1, 7), and C=(-2, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "67.62°", "metadata": {"A": ["4", "-5"], "B": ["-1", "7"], "C": ["-2", "6"], "angle_ABC_degrees": 67.61986494804043, "source_dataset": "advanced_geometry", "source_index": 151, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-8, 8), B=(-6, 1), and C=(2, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "126.50°", "metadata": {"A": ["-8", "8"], "B": ["-6", "1"], "C": ["2", "-2"], "angle_ABC_degrees": 126.50144112050633, "source_dataset": "advanced_geometry", "source_index": 152, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -9), B=(10, -4), and C=(-4, 2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "55.20°", "metadata": {"A": ["2", "-9"], "B": ["10", "-4"], "C": ["-4", "2"], "angle_ABC_degrees": 55.203973721731685, "source_dataset": "advanced_geometry", "source_index": 153, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 10), B=(-10, -4), and C=(3, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "5.94°", "metadata": {"A": ["6", "10"], "B": ["-10", "-4"], "C": ["3", "10"], "angle_ABC_degrees": 5.935171230951807, "source_dataset": "advanced_geometry", "source_index": 154, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, -6), B=(-5, -10), and C=(-4, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "54.52°", "metadata": {"A": ["-10", "-6"], "B": ["-5", "-10"], "C": ["-4", "8"], "angle_ABC_degrees": 54.52002186577415, "source_dataset": "advanced_geometry", "source_index": 155, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, 5), B=(-7, 5), and C=(4, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.000, 1.857)", "metadata": {"A": ["8", "5"], "B": ["-7", "5"], "C": ["4", "-9"], "ortho": ["4", "13/7"], "orthocenter_approx": [4.0, 1.8571428571428572], "source_dataset": "advanced_geometry", "source_index": 156, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, -2), B=(-3, 8), and C=(0, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.131", "metadata": {"A": ["1", "-2"], "B": ["-3", "8"], "C": ["0", "7"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(82)/2 + sqrt(29))*sqrt(-sqrt(29) + sqrt(10)/2 + sqrt(82)/2)*sqrt(-sqrt(82)/2 + sqrt(10)/2 + sqrt(29))/sqrt(sqrt(10)/2 + sqrt(82)/2 + sqrt(29))", "incircle_radius_approx": 1.1310252558539031, "source_dataset": "advanced_geometry", "source_index": 157, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -10), B=(-6, 9), and C=(-2, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.000, -10.421)", "metadata": {"A": ["-4", "-10"], "B": ["-6", "9"], "C": ["-2", "-10"], "ortho": ["-6", "-198/19"], "orthocenter_approx": [-6.0, -10.421052631578947], "source_dataset": "advanced_geometry", "source_index": 158, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 3), B=(-10, -1), and C=(0, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.400, -1.000)", "metadata": {"A": ["0", "3"], "B": ["-10", "-1"], "C": ["0", "5"], "ortho": ["12/5", "-1"], "orthocenter_approx": [2.4, -1.0], "source_dataset": "advanced_geometry", "source_index": 159, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, 5), B=(3, -4), and C=(-4, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.005", "metadata": {"A": ["10", "5"], "B": ["3", "-4"], "C": ["-4", "2"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + sqrt(85)/2 + sqrt(205)/2)*sqrt(-sqrt(205)/2 + sqrt(85)/2 + sqrt(130)/2)*sqrt(-sqrt(85)/2 + sqrt(130)/2 + sqrt(205)/2)/sqrt(sqrt(85)/2 + sqrt(130)/2 + sqrt(205)/2)", "incircle_radius_approx": 3.005227398014383, "source_dataset": "advanced_geometry", "source_index": 160, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, 5), B=(9, -4), and C=(-10, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-20.143, -11.286)", "metadata": {"A": ["-9", "5"], "B": ["9", "-4"], "C": ["-10", "9"], "ortho": ["-141/7", "-79/7"], "orthocenter_approx": [-20.142857142857142, -11.285714285714286], "source_dataset": "advanced_geometry", "source_index": 161, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 8), B=(-4, -6), and C=(3, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.641", "metadata": {"A": ["-8", "8"], "B": ["-4", "-6"], "C": ["3", "5"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + sqrt(170)/2 + sqrt(53))*sqrt(-sqrt(170)/2 + sqrt(130)/2 + sqrt(53))*sqrt(-sqrt(53) + sqrt(130)/2 + sqrt(170)/2)/sqrt(sqrt(130)/2 + sqrt(170)/2 + sqrt(53))", "incircle_radius_approx": 3.640990273010229, "source_dataset": "advanced_geometry", "source_index": 162, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, -7), B=(7, 5), and C=(4, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.805", "metadata": {"A": ["5", "-7"], "B": ["7", "5"], "C": ["4", "-3"], "incircle_radius_exact": "sqrt(-sqrt(17)/2 + sqrt(73)/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(17)/2 + sqrt(73)/2)*sqrt(-sqrt(73)/2 + sqrt(17)/2 + sqrt(37))/sqrt(sqrt(17)/2 + sqrt(73)/2 + sqrt(37))", "incircle_radius_approx": 0.8053917942191701, "source_dataset": "advanced_geometry", "source_index": 163, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, -1), B=(-3, -6), and C=(-3, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.000, -1.000)", "metadata": {"A": ["-8", "-1"], "B": ["-3", "-6"], "C": ["-3", "-3"], "ortho": ["-1", "-1"], "orthocenter_approx": [-1.0, -1.0], "source_dataset": "advanced_geometry", "source_index": 164, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -3), B=(-5, -10), and C=(1, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "7.13°", "metadata": {"A": ["9", "-3"], "B": ["-5", "-10"], "C": ["1", "-6"], "angle_ABC_degrees": 7.125016348901798, "source_dataset": "advanced_geometry", "source_index": 165, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -4), B=(-9, 8), and C=(8, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.293", "metadata": {"A": ["6", "-4"], "B": ["-9", "8"], "C": ["8", "-2"], "incircle_radius_exact": "sqrt(-sqrt(2) + 3*sqrt(41)/2 + sqrt(389)/2)*sqrt(-sqrt(389)/2 + sqrt(2) + 3*sqrt(41)/2)*sqrt(-3*sqrt(41)/2 + sqrt(2) + sqrt(389)/2)/sqrt(sqrt(2) + 3*sqrt(41)/2 + sqrt(389)/2)", "incircle_radius_approx": 1.2930761140723912, "source_dataset": "advanced_geometry", "source_index": 166, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, -5), B=(8, -1), and C=(-8, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "47.04°", "metadata": {"A": ["-10", "-5"], "B": ["8", "-1"], "C": ["-8", "10"], "angle_ABC_degrees": 47.037330696819915, "source_dataset": "advanced_geometry", "source_index": 167, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, 2), B=(-5, -10), and C=(10, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.436", "metadata": {"A": ["-10", "2"], "B": ["-5", "-10"], "C": ["10", "-1"], "incircle_radius_exact": "sqrt(-13/2 + 3*sqrt(34)/2 + sqrt(409)/2)*sqrt(-3*sqrt(34)/2 + 13/2 + sqrt(409)/2)*sqrt(-sqrt(409)/2 + 13/2 + 3*sqrt(34)/2)/sqrt(13/2 + 3*sqrt(34)/2 + sqrt(409)/2)", "incircle_radius_approx": 4.4364169090123955, "source_dataset": "advanced_geometry", "source_index": 168, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, -3), B=(-10, 0), and C=(10, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "34.55°", "metadata": {"A": ["1", "-3"], "B": ["-10", "0"], "C": ["10", "7"], "angle_ABC_degrees": 34.54516492224651, "source_dataset": "advanced_geometry", "source_index": 169, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, -9), B=(8, -9), and C=(-1, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.419", "metadata": {"A": ["9", "-9"], "B": ["8", "-9"], "C": ["-1", "7"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(337)/2 + sqrt(89))*sqrt(-sqrt(337)/2 + 1/2 + sqrt(89))*sqrt(-sqrt(89) + 1/2 + sqrt(337)/2)/sqrt(1/2 + sqrt(337)/2 + sqrt(89))", "incircle_radius_approx": 0.41856851539674444, "source_dataset": "advanced_geometry", "source_index": 170, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, -3), B=(-1, -10), and C=(-4, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.493, -2.425)", "metadata": {"A": ["3", "-3"], "B": ["-1", "-10"], "C": ["-4", "3"], "ortho": ["401/73", "-177/73"], "orthocenter_approx": [5.493150684931507, -2.4246575342465753], "source_dataset": "advanced_geometry", "source_index": 171, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, 3), B=(-1, 10), and C=(7, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "37.18°", "metadata": {"A": ["1", "3"], "B": ["-1", "10"], "C": ["7", "4"], "angle_ABC_degrees": 37.18470645323312, "source_dataset": "advanced_geometry", "source_index": 172, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, 0), B=(-7, -1), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.033, -7.447)", "metadata": {"A": ["10", "0"], "B": ["-7", "-1"], "C": ["-3", "-8"], "ortho": ["-373/123", "-916/123"], "orthocenter_approx": [-3.032520325203252, -7.4471544715447155], "source_dataset": "advanced_geometry", "source_index": 173, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -5), B=(5, 5), and C=(-10, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.909, 10.455)", "metadata": {"A": ["10", "-5"], "B": ["5", "5"], "C": ["-10", "2"], "ortho": ["76/11", "115/11"], "orthocenter_approx": [6.909090909090909, 10.454545454545455], "source_dataset": "advanced_geometry", "source_index": 174, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, -5), B=(-2, -3), and C=(-7, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.898", "metadata": {"A": ["-3", "-5"], "B": ["-2", "-3"], "C": ["-7", "0"], "incircle_radius_exact": "sqrt(-sqrt(34)/2 + sqrt(5)/2 + sqrt(41)/2)*sqrt(-sqrt(41)/2 + sqrt(5)/2 + sqrt(34)/2)*sqrt(-sqrt(5)/2 + sqrt(34)/2 + sqrt(41)/2)/sqrt(sqrt(5)/2 + sqrt(34)/2 + sqrt(41)/2)", "incircle_radius_approx": 0.8984015571217072, "source_dataset": "advanced_geometry", "source_index": 175, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, -7), B=(9, 1), and C=(0, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.062", "metadata": {"A": ["-5", "-7"], "B": ["9", "1"], "C": ["0", "-4"], "incircle_radius_exact": "sqrt(-sqrt(106)/2 + sqrt(34)/2 + sqrt(65))*sqrt(-sqrt(34)/2 + sqrt(106)/2 + sqrt(65))*sqrt(-sqrt(65) + sqrt(34)/2 + sqrt(106)/2)/sqrt(sqrt(34)/2 + sqrt(106)/2 + sqrt(65))", "incircle_radius_approx": 0.06201339343533791, "source_dataset": "advanced_geometry", "source_index": 176, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, 0), B=(6, -9), and C=(0, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "95.53°", "metadata": {"A": ["10", "0"], "B": ["6", "-9"], "C": ["0", "-7"], "angle_ABC_degrees": 95.52754015165618, "source_dataset": "advanced_geometry", "source_index": 177, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -3), B=(4, -2), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "76.33°", "metadata": {"A": ["-3", "-3"], "B": ["4", "-2"], "C": ["2", "3"], "angle_ABC_degrees": 76.32869286780416, "source_dataset": "advanced_geometry", "source_index": 178, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, -8), B=(5, -2), and C=(7, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.714, -2.571)", "metadata": {"A": ["3", "-8"], "B": ["5", "-2"], "C": ["7", "-3"], "ortho": ["40/7", "-18/7"], "orthocenter_approx": [5.714285714285714, -2.5714285714285716], "source_dataset": "advanced_geometry", "source_index": 179, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, -1), B=(-1, -9), and C=(-10, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(16.727, 11.682)", "metadata": {"A": ["-3", "-1"], "B": ["-1", "-9"], "C": ["-10", "5"], "ortho": ["184/11", "257/22"], "orthocenter_approx": [16.727272727272727, 11.681818181818182], "source_dataset": "advanced_geometry", "source_index": 180, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 4), B=(-4, 2), and C=(7, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.655", "metadata": {"A": ["1", "4"], "B": ["-4", "2"], "C": ["7", "10"], "incircle_radius_exact": "sqrt(-sqrt(185)/2 + sqrt(29)/2 + 3*sqrt(2))*sqrt(-3*sqrt(2) + sqrt(29)/2 + sqrt(185)/2)*sqrt(-sqrt(29)/2 + 3*sqrt(2) + sqrt(185)/2)/sqrt(sqrt(29)/2 + 3*sqrt(2) + sqrt(185)/2)", "incircle_radius_approx": 0.6552145670447903, "source_dataset": "advanced_geometry", "source_index": 181, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 4), B=(-8, 7), and C=(-3, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(8.634, 8.512)", "metadata": {"A": ["-4", "4"], "B": ["-8", "7"], "C": ["-3", "-7"], "ortho": ["354/41", "349/41"], "orthocenter_approx": [8.634146341463415, 8.512195121951219], "source_dataset": "advanced_geometry", "source_index": 182, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-3, -5), B=(9, -6), and C=(-4, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "23.54°", "metadata": {"A": ["-3", "-5"], "B": ["9", "-6"], "C": ["-4", "1"], "angle_ABC_degrees": 23.5371140752802, "source_dataset": "advanced_geometry", "source_index": 183, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, 10), B=(7, 8), and C=(9, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.511", "metadata": {"A": ["-9", "10"], "B": ["7", "8"], "C": ["9", "-9"], "incircle_radius_exact": "sqrt(-sqrt(293)/2 + sqrt(65) + sqrt(685)/2)*sqrt(-sqrt(65) + sqrt(293)/2 + sqrt(685)/2)*sqrt(-sqrt(685)/2 + sqrt(65) + sqrt(293)/2)/sqrt(sqrt(65) + sqrt(293)/2 + sqrt(685)/2)", "incircle_radius_approx": 4.510701417816483, "source_dataset": "advanced_geometry", "source_index": 184, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, -8), B=(4, -3), and C=(3, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.996", "metadata": {"A": ["1", "-8"], "B": ["4", "-3"], "C": ["3", "3"], "incircle_radius_exact": "sqrt(-sqrt(34)/2 + sqrt(37)/2 + 5*sqrt(5)/2)*sqrt(-sqrt(37)/2 + sqrt(34)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + sqrt(34)/2 + sqrt(37)/2)/sqrt(sqrt(34)/2 + sqrt(37)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 0.9959273364750433, "source_dataset": "advanced_geometry", "source_index": 185, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, 8), B=(-3, -3), and C=(4, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "32.47°", "metadata": {"A": ["-3", "8"], "B": ["-3", "-3"], "C": ["4", "8"], "angle_ABC_degrees": 32.47119229084849, "source_dataset": "advanced_geometry", "source_index": 186, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -3), B=(7, 3), and C=(-2, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "62.65°", "metadata": {"A": ["-6", "-3"], "B": ["7", "3"], "C": ["-2", "10"], "angle_ABC_degrees": 62.65012421993013, "source_dataset": "advanced_geometry", "source_index": 187, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, -1), B=(7, -10), and C=(3, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "36.87°", "metadata": {"A": ["-5", "-1"], "B": ["7", "-10"], "C": ["3", "-10"], "angle_ABC_degrees": 36.86989764584402, "source_dataset": "advanced_geometry", "source_index": 188, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, 2), B=(0, 8), and C=(-3, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-10.000, 3.000)", "metadata": {"A": ["-6", "2"], "B": ["0", "8"], "C": ["-3", "-4"], "ortho": ["-10", "3"], "orthocenter_approx": [-10.0, 3.0], "source_dataset": "advanced_geometry", "source_index": 189, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-7, -3), B=(-8, -8), and C=(-2, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.285", "metadata": {"A": ["-7", "-3"], "B": ["-8", "-8"], "C": ["-2", "-2"], "incircle_radius_exact": "3*sqrt(2)*sqrt(-3*sqrt(2) + sqrt(26))/sqrt(3*sqrt(2) + sqrt(26))", "incircle_radius_approx": 1.2845682397102496, "source_dataset": "advanced_geometry", "source_index": 190, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, -5), B=(6, -3), and C=(5, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "122.28°", "metadata": {"A": ["10", "-5"], "B": ["6", "-3"], "C": ["5", "7"], "angle_ABC_degrees": 122.27564431457762, "source_dataset": "advanced_geometry", "source_index": 191, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, -9), B=(8, 3), and C=(6, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.273", "metadata": {"A": ["-2", "-9"], "B": ["8", "3"], "C": ["6", "5"], "incircle_radius_exact": "sqrt(-sqrt(2) + sqrt(61) + sqrt(65))*sqrt(-sqrt(61) + sqrt(2) + sqrt(65))*sqrt(-sqrt(65) + sqrt(2) + sqrt(61))/sqrt(sqrt(2) + sqrt(61) + sqrt(65))", "incircle_radius_approx": 1.2726531548164148, "source_dataset": "advanced_geometry", "source_index": 192, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -9), B=(-3, -8), and C=(-8, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.808", "metadata": {"A": ["6", "-9"], "B": ["-3", "-8"], "C": ["-8", "-10"], "incircle_radius_exact": "sqrt(-sqrt(197)/2 + sqrt(29)/2 + sqrt(82)/2)*sqrt(-sqrt(29)/2 + sqrt(82)/2 + sqrt(197)/2)*sqrt(-sqrt(82)/2 + sqrt(29)/2 + sqrt(197)/2)/sqrt(sqrt(29)/2 + sqrt(82)/2 + sqrt(197)/2)", "incircle_radius_approx": 0.8076915045245611, "source_dataset": "advanced_geometry", "source_index": 193, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 5), B=(1, -8), and C=(2, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.57°", "metadata": {"A": ["1", "5"], "B": ["1", "-8"], "C": ["2", "-6"], "angle_ABC_degrees": 26.56505117707799, "source_dataset": "advanced_geometry", "source_index": 194, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, -5), B=(-3, 3), and C=(-9, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "81.87°", "metadata": {"A": ["5", "-5"], "B": ["-3", "3"], "C": ["-9", "-5"], "angle_ABC_degrees": 81.86989764584402, "source_dataset": "advanced_geometry", "source_index": 195, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, 3), B=(10, -8), and C=(-8, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.361", "metadata": {"A": ["-8", "3"], "B": ["10", "-8"], "C": ["-8", "-6"], "incircle_radius_exact": "sqrt(-9/2 + sqrt(82) + sqrt(445)/2)*sqrt(-sqrt(445)/2 + 9/2 + sqrt(82))*sqrt(-sqrt(82) + 9/2 + sqrt(445)/2)/sqrt(9/2 + sqrt(82) + sqrt(445)/2)", "incircle_radius_approx": 3.3605919251820766, "source_dataset": "advanced_geometry", "source_index": 196, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -6), B=(3, -7), and C=(3, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.429, -6.000)", "metadata": {"A": ["-4", "-6"], "B": ["3", "-7"], "C": ["3", "-9"], "ortho": ["24/7", "-6"], "orthocenter_approx": [3.4285714285714284, -6.0], "source_dataset": "advanced_geometry", "source_index": 197, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, -9), B=(0, -6), and C=(9, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "145.30°", "metadata": {"A": ["-9", "-9"], "B": ["0", "-6"], "C": ["9", "6"], "angle_ABC_degrees": 145.30484646876604, "source_dataset": "advanced_geometry", "source_index": 198, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(7, -7), B=(1, -8), and C=(-3, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "93.78°", "metadata": {"A": ["7", "-7"], "B": ["1", "-8"], "C": ["-3", "9"], "angle_ABC_degrees": 93.77819770716158, "source_dataset": "advanced_geometry", "source_index": 199, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, 5), B=(6, 9), and C=(5, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.268", "metadata": {"A": ["-10", "5"], "B": ["6", "9"], "C": ["5", "6"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(226)/2 + 2*sqrt(17))*sqrt(-2*sqrt(17) + sqrt(10)/2 + sqrt(226)/2)*sqrt(-sqrt(226)/2 + sqrt(10)/2 + 2*sqrt(17))/sqrt(sqrt(10)/2 + sqrt(226)/2 + 2*sqrt(17))", "incircle_radius_approx": 1.2684503109880794, "source_dataset": "advanced_geometry", "source_index": 200, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, 3), B=(-9, 4), and C=(8, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "10.05°", "metadata": {"A": ["0", "3"], "B": ["-9", "4"], "C": ["8", "-1"], "angle_ABC_degrees": 10.049348588124873, "source_dataset": "advanced_geometry", "source_index": 201, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(3, 2), B=(8, 4), and C=(-8, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "11.18°", "metadata": {"A": ["3", "2"], "B": ["8", "4"], "C": ["-8", "1"], "angle_ABC_degrees": 11.181754210196678, "source_dataset": "advanced_geometry", "source_index": 202, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, 5), B=(-8, 4), and C=(0, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.037, 1.741)", "metadata": {"A": ["-1", "5"], "B": ["-8", "4"], "C": ["0", "9"], "ortho": ["28/27", "47/27"], "orthocenter_approx": [1.037037037037037, 1.7407407407407407], "source_dataset": "advanced_geometry", "source_index": 203, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, 2), B=(1, -8), and C=(4, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(22.000, -14.000)", "metadata": {"A": ["6", "2"], "B": ["1", "-8"], "C": ["4", "-5"], "ortho": ["22", "-14"], "orthocenter_approx": [22.0, -14.0], "source_dataset": "advanced_geometry", "source_index": 204, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -10), B=(1, 5), and C=(-9, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.289", "metadata": {"A": ["0", "-10"], "B": ["1", "5"], "C": ["-9", "9"], "incircle_radius_exact": "sqrt(-sqrt(226)/2 + sqrt(29) + sqrt(442)/2)*sqrt(-sqrt(29) + sqrt(226)/2 + sqrt(442)/2)*sqrt(-sqrt(442)/2 + sqrt(29) + sqrt(226)/2)/sqrt(sqrt(29) + sqrt(226)/2 + sqrt(442)/2)", "incircle_radius_approx": 3.2886713235525846, "source_dataset": "advanced_geometry", "source_index": 205, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, -4), B=(4, -7), and C=(0, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.137", "metadata": {"A": ["5", "-4"], "B": ["4", "-7"], "C": ["0", "-2"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(29)/2 + sqrt(41)/2)*sqrt(-sqrt(29)/2 + sqrt(10)/2 + sqrt(41)/2)*sqrt(-sqrt(41)/2 + sqrt(10)/2 + sqrt(29)/2)/sqrt(sqrt(10)/2 + sqrt(29)/2 + sqrt(41)/2)", "incircle_radius_approx": 1.1370806428772422, "source_dataset": "advanced_geometry", "source_index": 206, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, 10), B=(5, 7), and C=(-5, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.088", "metadata": {"A": ["10", "10"], "B": ["5", "7"], "C": ["-5", "-10"], "incircle_radius_exact": "sqrt(-25/2 + sqrt(34)/2 + sqrt(389)/2)*sqrt(-sqrt(34)/2 + sqrt(389)/2 + 25/2)*sqrt(-sqrt(389)/2 + sqrt(34)/2 + 25/2)/sqrt(sqrt(34)/2 + sqrt(389)/2 + 25/2)", "incircle_radius_approx": 1.087944813857696, "source_dataset": "advanced_geometry", "source_index": 207, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -5), B=(7, -8), and C=(-9, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.613", "metadata": {"A": ["2", "-5"], "B": ["7", "-8"], "C": ["-9", "7"], "incircle_radius_exact": "sqrt(-sqrt(265)/2 + sqrt(34)/2 + sqrt(481)/2)*sqrt(-sqrt(34)/2 + sqrt(265)/2 + sqrt(481)/2)*sqrt(-sqrt(481)/2 + sqrt(34)/2 + sqrt(265)/2)/sqrt(sqrt(34)/2 + sqrt(265)/2 + sqrt(481)/2)", "incircle_radius_approx": 0.6130583514565641, "source_dataset": "advanced_geometry", "source_index": 208, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 8), B=(8, 4), and C=(-3, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.974", "metadata": {"A": ["2", "8"], "B": ["8", "4"], "C": ["-3", "3"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + 5*sqrt(2)/2 + sqrt(13))*sqrt(-sqrt(13) + 5*sqrt(2)/2 + sqrt(122)/2)*sqrt(-5*sqrt(2)/2 + sqrt(13) + sqrt(122)/2)/sqrt(5*sqrt(2)/2 + sqrt(13) + sqrt(122)/2)", "incircle_radius_approx": 1.9741363360631663, "source_dataset": "advanced_geometry", "source_index": 209, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, -1), B=(7, -6), and C=(-4, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "12.22°", "metadata": {"A": ["-10", "-1"], "B": ["7", "-6"], "C": ["-4", "0"], "angle_ABC_degrees": 12.220919331930434, "source_dataset": "advanced_geometry", "source_index": 210, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(7, 2), B=(3, -6), and C=(10, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.94°", "metadata": {"A": ["7", "2"], "B": ["3", "-6"], "C": ["10", "10"], "angle_ABC_degrees": 2.9356734464211725, "source_dataset": "advanced_geometry", "source_index": 211, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -2), B=(3, 0), and C=(-4, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.537", "metadata": {"A": ["6", "-2"], "B": ["3", "0"], "C": ["-4", "-10"], "incircle_radius_exact": "sqrt(-sqrt(13)/2 + sqrt(149)/2 + sqrt(41))*sqrt(-sqrt(149)/2 + sqrt(13)/2 + sqrt(41))*sqrt(-sqrt(41) + sqrt(13)/2 + sqrt(149)/2)/sqrt(sqrt(13)/2 + sqrt(149)/2 + sqrt(41))", "incircle_radius_approx": 1.5374747932642099, "source_dataset": "advanced_geometry", "source_index": 212, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 10), B=(10, 8), and C=(-9, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.667, 22.667)", "metadata": {"A": ["0", "10"], "B": ["10", "8"], "C": ["-9", "1"], "ortho": ["-14/3", "68/3"], "orthocenter_approx": [-4.666666666666667, 22.666666666666668], "source_dataset": "advanced_geometry", "source_index": 213, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, 3), B=(1, 2), and C=(-9, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.293", "metadata": {"A": ["-1", "3"], "B": ["1", "2"], "C": ["-9", "4"], "incircle_radius_exact": "sqrt(-sqrt(26) + sqrt(5)/2 + sqrt(65)/2)*sqrt(-sqrt(5)/2 + sqrt(65)/2 + sqrt(26))*sqrt(-sqrt(65)/2 + sqrt(5)/2 + sqrt(26))/sqrt(sqrt(5)/2 + sqrt(65)/2 + sqrt(26))", "incircle_radius_approx": 0.29273483724115057, "source_dataset": "advanced_geometry", "source_index": 214, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -7), B=(-2, 7), and C=(7, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.225", "metadata": {"A": ["3", "-7"], "B": ["-2", "7"], "C": ["7", "6"], "incircle_radius_exact": "sqrt(-sqrt(185)/2 + sqrt(82)/2 + sqrt(221)/2)*sqrt(-sqrt(221)/2 + sqrt(82)/2 + sqrt(185)/2)*sqrt(-sqrt(82)/2 + sqrt(185)/2 + sqrt(221)/2)/sqrt(sqrt(82)/2 + sqrt(185)/2 + sqrt(221)/2)", "incircle_radius_approx": 3.2246953550009305, "source_dataset": "advanced_geometry", "source_index": 215, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, -6), B=(7, 3), and C=(-4, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.33°", "metadata": {"A": ["-8", "-6"], "B": ["7", "3"], "C": ["-4", "-6"], "angle_ABC_degrees": 8.325650330426836, "source_dataset": "advanced_geometry", "source_index": 216, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(7, -8), B=(3, 9), and C=(5, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.263, 8.526)", "metadata": {"A": ["7", "-8"], "B": ["3", "9"], "C": ["5", "10"], "ortho": ["-24/19", "162/19"], "orthocenter_approx": [-1.263157894736842, 8.526315789473685], "source_dataset": "advanced_geometry", "source_index": 217, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 3), B=(8, 8), and C=(5, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.729, 0.068)", "metadata": {"A": ["-8", "3"], "B": ["8", "8"], "C": ["5", "-4"], "ortho": ["220/59", "4/59"], "orthocenter_approx": [3.7288135593220337, 0.06779661016949153], "source_dataset": "advanced_geometry", "source_index": 218, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -5), B=(-9, 7), and C=(8, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.301", "metadata": {"A": ["-10", "-5"], "B": ["-9", "7"], "C": ["8", "-2"], "incircle_radius_exact": "sqrt(-sqrt(145)/2 + 3*sqrt(37)/2 + sqrt(370)/2)*sqrt(-3*sqrt(37)/2 + sqrt(145)/2 + sqrt(370)/2)*sqrt(-sqrt(370)/2 + sqrt(145)/2 + 3*sqrt(37)/2)/sqrt(sqrt(145)/2 + 3*sqrt(37)/2 + sqrt(370)/2)", "incircle_radius_approx": 4.300835032465371, "source_dataset": "advanced_geometry", "source_index": 219, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, 3), B=(4, 0), and C=(1, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.645", "metadata": {"A": ["7", "3"], "B": ["4", "0"], "C": ["1", "0"], "incircle_radius_exact": "sqrt(-3/2 + 3*sqrt(2)/2 + 3*sqrt(5)/2)*sqrt(-3*sqrt(2)/2 + 3/2 + 3*sqrt(5)/2)*sqrt(-3*sqrt(5)/2 + 3/2 + 3*sqrt(2)/2)/sqrt(3/2 + 3*sqrt(2)/2 + 3*sqrt(5)/2)", "incircle_radius_approx": 0.6451222306170316, "source_dataset": "advanced_geometry", "source_index": 220, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, 0), B=(9, 2), and C=(-2, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "17.59°", "metadata": {"A": ["3", "0"], "B": ["9", "2"], "C": ["-2", "-6"], "angle_ABC_degrees": 17.5924245621816, "source_dataset": "advanced_geometry", "source_index": 221, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-1, 7), B=(-4, 6), and C=(-5, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "117.90°", "metadata": {"A": ["-1", "7"], "B": ["-4", "6"], "C": ["-5", "0"], "angle_ABC_degrees": 117.89727103094762, "source_dataset": "advanced_geometry", "source_index": 222, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 0), B=(5, 6), and C=(0, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.272", "metadata": {"A": ["5", "0"], "B": ["5", "6"], "C": ["0", "-4"], "incircle_radius_exact": "sqrt(-3 + sqrt(41)/2 + 5*sqrt(5)/2)*sqrt(-sqrt(41)/2 + 3 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + 3 + sqrt(41)/2)/sqrt(3 + sqrt(41)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 1.2720777507950927, "source_dataset": "advanced_geometry", "source_index": 223, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, -7), B=(4, -4), and C=(7, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, -4.000)", "metadata": {"A": ["7", "-7"], "B": ["4", "-4"], "C": ["7", "2"], "ortho": ["1", "-4"], "orthocenter_approx": [1.0, -4.0], "source_dataset": "advanced_geometry", "source_index": 224, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, 4), B=(0, -7), and C=(-10, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-25.400, -7.000)", "metadata": {"A": ["-10", "4"], "B": ["0", "-7"], "C": ["-10", "7"], "ortho": ["-127/5", "-7"], "orthocenter_approx": [-25.4, -7.0], "source_dataset": "advanced_geometry", "source_index": 225, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, 0), B=(-9, 3), and C=(-7, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "65.34°", "metadata": {"A": ["7", "0"], "B": ["-9", "3"], "C": ["-7", "-5"], "angle_ABC_degrees": 65.34410125591837, "source_dataset": "advanced_geometry", "source_index": 226, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, -8), B=(9, -1), and C=(-4, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.455, 12.091)", "metadata": {"A": ["-10", "-8"], "B": ["9", "-1"], "C": ["-4", "0"], "ortho": ["-93/11", "133/11"], "orthocenter_approx": [-8.454545454545455, 12.09090909090909], "source_dataset": "advanced_geometry", "source_index": 227, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, -8), B=(9, -1), and C=(-10, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.054, -5.776)", "metadata": {"A": ["1", "-8"], "B": ["9", "-1"], "C": ["-10", "8"], "ortho": ["421/205", "-1184/205"], "orthocenter_approx": [2.053658536585366, -5.775609756097561], "source_dataset": "advanced_geometry", "source_index": 228, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, 1), B=(0, -5), and C=(1, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.151", "metadata": {"A": ["10", "1"], "B": ["0", "-5"], "C": ["1", "6"], "incircle_radius_exact": "sqrt(-sqrt(106)/2 + sqrt(122)/2 + sqrt(34))*sqrt(-sqrt(122)/2 + sqrt(106)/2 + sqrt(34))*sqrt(-sqrt(34) + sqrt(106)/2 + sqrt(122)/2)/sqrt(sqrt(106)/2 + sqrt(122)/2 + sqrt(34))", "incircle_radius_approx": 3.151238706916174, "source_dataset": "advanced_geometry", "source_index": 229, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, -8), B=(10, 5), and C=(0, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.638", "metadata": {"A": ["1", "-8"], "B": ["10", "5"], "C": ["0", "1"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(82)/2 + sqrt(29))*sqrt(-sqrt(29) + sqrt(82)/2 + 5*sqrt(10)/2)*sqrt(-sqrt(82)/2 + sqrt(29) + 5*sqrt(10)/2)/sqrt(sqrt(82)/2 + sqrt(29) + 5*sqrt(10)/2)", "incircle_radius_approx": 2.637700372545627, "source_dataset": "advanced_geometry", "source_index": 230, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, 8), B=(-3, -1), and C=(9, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, 25.333)", "metadata": {"A": ["10", "8"], "B": ["-3", "-1"], "C": ["9", "8"], "ortho": ["-3", "76/3"], "orthocenter_approx": [-3.0, 25.333333333333332], "source_dataset": "advanced_geometry", "source_index": 231, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 1), B=(3, -9), and C=(-4, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.710, -10.971)", "metadata": {"A": ["4", "1"], "B": ["3", "-9"], "C": ["-4", "-10"], "ortho": ["394/69", "-757/69"], "orthocenter_approx": [5.7101449275362315, -10.971014492753623], "source_dataset": "advanced_geometry", "source_index": 232, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, 1), B=(7, -1), and C=(0, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(16.000, -13.000)", "metadata": {"A": ["8", "1"], "B": ["7", "-1"], "C": ["0", "-5"], "ortho": ["16", "-13"], "orthocenter_approx": [16.0, -13.0], "source_dataset": "advanced_geometry", "source_index": 233, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, -4), B=(0, -2), and C=(-2, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.402", "metadata": {"A": ["-8", "-4"], "B": ["0", "-2"], "C": ["-2", "-10"], "incircle_radius_exact": "3*sqrt(2)*sqrt(-3*sqrt(2) + 2*sqrt(17))/sqrt(3*sqrt(2) + 2*sqrt(17))", "incircle_radius_approx": 2.4021423384696217, "source_dataset": "advanced_geometry", "source_index": 234, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, -4), B=(-9, -5), and C=(5, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(12.789, -23.158)", "metadata": {"A": ["-5", "-4"], "B": ["-9", "-5"], "C": ["5", "8"], "ortho": ["243/19", "-440/19"], "orthocenter_approx": [12.789473684210526, -23.157894736842106], "source_dataset": "advanced_geometry", "source_index": 235, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, -2), B=(-9, 5), and C=(10, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.075", "metadata": {"A": ["5", "-2"], "B": ["-9", "5"], "C": ["10", "5"], "incircle_radius_exact": "sqrt(-19/2 + sqrt(74)/2 + 7*sqrt(5)/2)*sqrt(-7*sqrt(5)/2 + sqrt(74)/2 + 19/2)*sqrt(-sqrt(74)/2 + 7*sqrt(5)/2 + 19/2)/sqrt(sqrt(74)/2 + 7*sqrt(5)/2 + 19/2)", "incircle_radius_approx": 3.0748031799564286, "source_dataset": "advanced_geometry", "source_index": 236, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, -9), B=(5, -5), and C=(-5, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.170", "metadata": {"A": ["1", "-9"], "B": ["5", "-5"], "C": ["-5", "0"], "incircle_radius_exact": "sqrt(-3*sqrt(13)/2 + 2*sqrt(2) + 5*sqrt(5)/2)*sqrt(-2*sqrt(2) + 3*sqrt(13)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + 2*sqrt(2) + 3*sqrt(13)/2)/sqrt(2*sqrt(2) + 3*sqrt(13)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 2.1696799692920323, "source_dataset": "advanced_geometry", "source_index": 237, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, 10), B=(-8, 10), and C=(-2, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.485", "metadata": {"A": ["4", "10"], "B": ["-8", "10"], "C": ["-2", "4"], "incircle_radius_exact": "sqrt(-6 + 6*sqrt(2))*sqrt(216 + 216*sqrt(2))/(6 + 6*sqrt(2))", "incircle_radius_approx": 2.4852813742385704, "source_dataset": "advanced_geometry", "source_index": 238, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -2), B=(-7, 4), and C=(4, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.462, 0.538)", "metadata": {"A": ["-1", "-2"], "B": ["-7", "4"], "C": ["4", "6"], "ortho": ["-19/13", "7/13"], "orthocenter_approx": [-1.4615384615384615, 0.5384615384615384], "source_dataset": "advanced_geometry", "source_index": 239, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, 5), B=(2, 0), and C=(-6, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "65.63°", "metadata": {"A": ["-9", "5"], "B": ["2", "0"], "C": ["-6", "-7"], "angle_ABC_degrees": 65.62987994612618, "source_dataset": "advanced_geometry", "source_index": 240, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, -6), B=(6, -2), and C=(4, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "94.57°", "metadata": {"A": ["-10", "-6"], "B": ["6", "-2"], "C": ["4", "10"], "angle_ABC_degrees": 94.57392125990086, "source_dataset": "advanced_geometry", "source_index": 241, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-2, 1), B=(-7, 4), and C=(-1, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.714, 3.143)", "metadata": {"A": ["-2", "1"], "B": ["-7", "4"], "C": ["-1", "6"], "ortho": ["-19/7", "22/7"], "orthocenter_approx": [-2.7142857142857144, 3.142857142857143], "source_dataset": "advanced_geometry", "source_index": 242, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 2), B=(8, 8), and C=(4, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "42.27°", "metadata": {"A": ["-1", "2"], "B": ["8", "8"], "C": ["4", "-8"], "angle_ABC_degrees": 42.27368900609373, "source_dataset": "advanced_geometry", "source_index": 243, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 0), B=(-8, -8), and C=(3, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.517", "metadata": {"A": ["-8", "0"], "B": ["-8", "-8"], "C": ["3", "3"], "incircle_radius_exact": "sqrt(-4 + sqrt(130)/2 + 11*sqrt(2)/2)*sqrt(-sqrt(130)/2 + 4 + 11*sqrt(2)/2)*sqrt(-11*sqrt(2)/2 + 4 + sqrt(130)/2)/sqrt(4 + sqrt(130)/2 + 11*sqrt(2)/2)", "incircle_radius_approx": 2.5172990336374976, "source_dataset": "advanced_geometry", "source_index": 244, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, 0), B=(-2, -7), and C=(-5, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "32.11°", "metadata": {"A": ["-10", "0"], "B": ["-2", "-7"], "C": ["-5", "3"], "angle_ABC_degrees": 32.114830600296735, "source_dataset": "advanced_geometry", "source_index": 245, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 5), B=(5, -9), and C=(6, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.495", "metadata": {"A": ["5", "5"], "B": ["5", "-9"], "C": ["6", "3"], "incircle_radius_exact": "sqrt(-7 + sqrt(5)/2 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + sqrt(5)/2 + 7)*sqrt(-sqrt(5)/2 + sqrt(145)/2 + 7)/sqrt(sqrt(5)/2 + sqrt(145)/2 + 7)", "incircle_radius_approx": 0.4950904259547736, "source_dataset": "advanced_geometry", "source_index": 246, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, -8), B=(4, -10), and C=(-10, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.060", "metadata": {"A": ["-5", "-8"], "B": ["4", "-10"], "C": ["-10", "2"], "incircle_radius_exact": "sqrt(-5*sqrt(5)/2 + 3*sqrt(85)/2)*sqrt(-sqrt(85)/2 + 5*sqrt(5)/2)*sqrt(sqrt(85)/2 + 5*sqrt(5)/2)/sqrt(5*sqrt(5)/2 + 3*sqrt(85)/2)", "incircle_radius_approx": 2.059786685547464, "source_dataset": "advanced_geometry", "source_index": 247, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, 5), B=(8, -10), and C=(-6, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "35.36°", "metadata": {"A": ["0", "5"], "B": ["8", "-10"], "C": ["-6", "-3"], "angle_ABC_degrees": 35.36246188706905, "source_dataset": "advanced_geometry", "source_index": 248, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 10), B=(6, -8), and C=(2, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "9.35°", "metadata": {"A": ["5", "10"], "B": ["6", "-8"], "C": ["2", "10"], "angle_ABC_degrees": 9.348977589287278, "source_dataset": "advanced_geometry", "source_index": 249, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, -6), B=(0, -4), and C=(7, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-217.000, 120.000)", "metadata": {"A": ["-1", "-6"], "B": ["0", "-4"], "C": ["7", "8"], "ortho": ["-217", "120"], "orthocenter_approx": [-217.0, 120.0], "source_dataset": "advanced_geometry", "source_index": 250, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, -7), B=(-3, 6), and C=(6, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.672", "metadata": {"A": ["5", "-7"], "B": ["-3", "6"], "C": ["6", "3"], "incircle_radius_exact": "sqrt(-3*sqrt(10)/2 + sqrt(101)/2 + sqrt(233)/2)*sqrt(-sqrt(101)/2 + 3*sqrt(10)/2 + sqrt(233)/2)*sqrt(-sqrt(233)/2 + 3*sqrt(10)/2 + sqrt(101)/2)/sqrt(3*sqrt(10)/2 + sqrt(101)/2 + sqrt(233)/2)", "incircle_radius_approx": 2.672333459987496, "source_dataset": "advanced_geometry", "source_index": 251, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, 5), B=(-4, -8), and C=(3, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.227, 4.682)", "metadata": {"A": ["-9", "5"], "B": ["-4", "-8"], "C": ["3", "9"], "ortho": ["-181/22", "103/22"], "orthocenter_approx": [-8.227272727272727, 4.681818181818182], "source_dataset": "advanced_geometry", "source_index": 252, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 0), B=(8, 5), and C=(-8, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.619", "metadata": {"A": ["5", "0"], "B": ["8", "5"], "C": ["-8", "-2"], "incircle_radius_exact": "sqrt(-sqrt(173)/2 + sqrt(34)/2 + sqrt(305)/2)*sqrt(-sqrt(305)/2 + sqrt(34)/2 + sqrt(173)/2)*sqrt(-sqrt(34)/2 + sqrt(173)/2 + sqrt(305)/2)/sqrt(sqrt(34)/2 + sqrt(173)/2 + sqrt(305)/2)", "incircle_radius_approx": 1.618737960617469, "source_dataset": "advanced_geometry", "source_index": 253, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, 1), B=(6, -9), and C=(10, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.511, 0.409)", "metadata": {"A": ["-2", "1"], "B": ["6", "-9"], "C": ["10", "8"], "ortho": ["45/88", "9/22"], "orthocenter_approx": [0.5113636363636364, 0.40909090909090906], "source_dataset": "advanced_geometry", "source_index": 254, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, 2), B=(-9, 0), and C=(-7, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "85.36°", "metadata": {"A": ["1", "2"], "B": ["-9", "0"], "C": ["-7", "-7"], "angle_ABC_degrees": 85.36453657309737, "source_dataset": "advanced_geometry", "source_index": 255, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 10), B=(6, -5), and C=(9, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.000, -4.867)", "metadata": {"A": ["7", "10"], "B": ["6", "-5"], "C": ["9", "-5"], "ortho": ["7", "-73/15"], "orthocenter_approx": [7.0, -4.866666666666666], "source_dataset": "advanced_geometry", "source_index": 256, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, -4), B=(-10, 4), and C=(6, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.136", "metadata": {"A": ["8", "-4"], "B": ["-10", "4"], "C": ["6", "8"], "incircle_radius_exact": "sqrt(-2*sqrt(17) + sqrt(37) + sqrt(97))*sqrt(-sqrt(37) + 2*sqrt(17) + sqrt(97))*sqrt(-sqrt(97) + sqrt(37) + 2*sqrt(17))/sqrt(sqrt(37) + 2*sqrt(17) + sqrt(97))", "incircle_radius_approx": 4.136020207409705, "source_dataset": "advanced_geometry", "source_index": 257, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 1), B=(-10, 5), and C=(1, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "25.18°", "metadata": {"A": ["1", "1"], "B": ["-10", "5"], "C": ["1", "6"], "angle_ABC_degrees": 25.17753542963478, "source_dataset": "advanced_geometry", "source_index": 258, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, 5), B=(7, 4), and C=(10, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.278", "metadata": {"A": ["10", "5"], "B": ["7", "4"], "C": ["10", "-3"], "incircle_radius_exact": "sqrt(-4 + sqrt(10)/2 + sqrt(58)/2)*sqrt(-sqrt(10)/2 + sqrt(58)/2 + 4)*sqrt(-sqrt(58)/2 + sqrt(10)/2 + 4)/sqrt(sqrt(10)/2 + sqrt(58)/2 + 4)", "incircle_radius_approx": 1.2780879282429956, "source_dataset": "advanced_geometry", "source_index": 259, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, 7), B=(3, 5), and C=(-4, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "143.13°", "metadata": {"A": ["7", "7"], "B": ["3", "5"], "C": ["-4", "-9"], "angle_ABC_degrees": 143.13010235415598, "source_dataset": "advanced_geometry", "source_index": 260, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-3, -4), B=(-8, 8), and C=(9, -2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.338, -7.974)", "metadata": {"A": ["-3", "-4"], "B": ["-8", "8"], "C": ["9", "-2"], "ortho": ["-411/77", "-614/77"], "orthocenter_approx": [-5.337662337662338, -7.974025974025974], "source_dataset": "advanced_geometry", "source_index": 261, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 3), B=(6, 4), and C=(-9, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-11.127, 18.273)", "metadata": {"A": ["-4", "3"], "B": ["6", "4"], "C": ["-9", "-3"], "ortho": ["-612/55", "201/11"], "orthocenter_approx": [-11.127272727272727, 18.272727272727273], "source_dataset": "advanced_geometry", "source_index": 262, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, 8), B=(2, 10), and C=(-1, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.333, 13.000)", "metadata": {"A": ["8", "8"], "B": ["2", "10"], "C": ["-1", "6"], "ortho": ["4/3", "13"], "orthocenter_approx": [1.3333333333333333, 13.0], "source_dataset": "advanced_geometry", "source_index": 263, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, -4), B=(-2, 9), and C=(-4, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.190", "metadata": {"A": ["6", "-4"], "B": ["-2", "9"], "C": ["-4", "3"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(149)/2 + sqrt(233)/2)*sqrt(-sqrt(149)/2 + sqrt(10) + sqrt(233)/2)*sqrt(-sqrt(233)/2 + sqrt(10) + sqrt(149)/2)/sqrt(sqrt(10) + sqrt(149)/2 + sqrt(233)/2)", "incircle_radius_approx": 2.189643972050657, "source_dataset": "advanced_geometry", "source_index": 264, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, -1), B=(-1, 6), and C=(5, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.43°", "metadata": {"A": ["6", "-1"], "B": ["-1", "6"], "C": ["5", "3"], "angle_ABC_degrees": 18.43494882292201, "source_dataset": "advanced_geometry", "source_index": 265, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-6, 2), B=(1, 1), and C=(6, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(10.500, 29.500)", "metadata": {"A": ["-6", "2"], "B": ["1", "1"], "C": ["6", "-2"], "ortho": ["21/2", "59/2"], "orthocenter_approx": [10.5, 29.5], "source_dataset": "advanced_geometry", "source_index": 266, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -10), B=(-6, -4), and C=(-3, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.000, -2.333)", "metadata": {"A": ["-1", "-10"], "B": ["-6", "-4"], "C": ["-3", "-4"], "ortho": ["-1", "-7/3"], "orthocenter_approx": [-1.0, -2.3333333333333335], "source_dataset": "advanced_geometry", "source_index": 267, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-8, -4), B=(-2, -5), and C=(-1, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "84.94°", "metadata": {"A": ["-8", "-4"], "B": ["-2", "-5"], "C": ["-1", "8"], "angle_ABC_degrees": 84.93638314696992, "source_dataset": "advanced_geometry", "source_index": 268, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, -8), B=(3, 5), and C=(-6, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.520", "metadata": {"A": ["-2", "-8"], "B": ["3", "5"], "C": ["-6", "-2"], "incircle_radius_exact": "sqrt(-sqrt(13) + sqrt(130)/2 + sqrt(194)/2)*sqrt(-sqrt(130)/2 + sqrt(13) + sqrt(194)/2)*sqrt(-sqrt(194)/2 + sqrt(13) + sqrt(130)/2)/sqrt(sqrt(13) + sqrt(130)/2 + sqrt(194)/2)", "incircle_radius_approx": 2.519878996660048, "source_dataset": "advanced_geometry", "source_index": 269, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 2), B=(-9, -6), and C=(0, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "51.01°", "metadata": {"A": ["-6", "2"], "B": ["-9", "-6"], "C": ["0", "-3"], "angle_ABC_degrees": 51.00900595749453, "source_dataset": "advanced_geometry", "source_index": 270, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, 3), B=(5, -10), and C=(-4, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "28.47°", "metadata": {"A": ["-5", "3"], "B": ["5", "-10"], "C": ["-4", "-6"], "angle_ABC_degrees": 28.468918996594322, "source_dataset": "advanced_geometry", "source_index": 271, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, -7), B=(-5, -10), and C=(10, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(19.556, -18.667)", "metadata": {"A": ["4", "-7"], "B": ["-5", "-10"], "C": ["10", "10"], "ortho": ["176/9", "-56/3"], "orthocenter_approx": [19.555555555555557, -18.666666666666668], "source_dataset": "advanced_geometry", "source_index": 272, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, 0), B=(-10, 1), and C=(8, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.044", "metadata": {"A": ["3", "0"], "B": ["-10", "1"], "C": ["8", "10"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + 7*sqrt(5))*sqrt(-2*sqrt(5) + sqrt(170)/2)*sqrt(2*sqrt(5) + sqrt(170)/2)/sqrt(sqrt(170)/2 + 7*sqrt(5))", "incircle_radius_approx": 3.0444244790986263, "source_dataset": "advanced_geometry", "source_index": 273, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 0), B=(-10, -3), and C=(-3, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-15.714, 5.000)", "metadata": {"A": ["-10", "0"], "B": ["-10", "-3"], "C": ["-3", "5"], "ortho": ["-110/7", "5"], "orthocenter_approx": [-15.714285714285714, 5.0], "source_dataset": "advanced_geometry", "source_index": 274, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 2), B=(-10, -4), and C=(-3, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.519", "metadata": {"A": ["-10", "2"], "B": ["-10", "-4"], "C": ["-3", "7"], "incircle_radius_exact": "sqrt(-3 + sqrt(74)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + 3 + sqrt(74)/2)*sqrt(-sqrt(74)/2 + 3 + sqrt(170)/2)/sqrt(3 + sqrt(74)/2 + sqrt(170)/2)", "incircle_radius_approx": 1.5194967673544848, "source_dataset": "advanced_geometry", "source_index": 275, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -9), B=(4, 2), and C=(3, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "36.74°", "metadata": {"A": ["10", "-9"], "B": ["4", "2"], "C": ["3", "-5"], "angle_ABC_degrees": 36.740562020121196, "source_dataset": "advanced_geometry", "source_index": 276, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 7), B=(9, 9), and C=(2, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.000, 108.000)", "metadata": {"A": ["-9", "7"], "B": ["9", "9"], "C": ["2", "9"], "ortho": ["-9", "108"], "orthocenter_approx": [-9.0, 108.0], "source_dataset": "advanced_geometry", "source_index": 277, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, 10), B=(-8, 4), and C=(7, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.578", "metadata": {"A": ["6", "10"], "B": ["-8", "4"], "C": ["7", "0"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(58) + sqrt(241)/2)*sqrt(-sqrt(241)/2 + sqrt(101)/2 + sqrt(58))*sqrt(-sqrt(58) + sqrt(101)/2 + sqrt(241)/2)/sqrt(sqrt(101)/2 + sqrt(58) + sqrt(241)/2)", "incircle_radius_approx": 3.577940587043022, "source_dataset": "advanced_geometry", "source_index": 278, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -6), B=(1, 7), and C=(4, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.186", "metadata": {"A": ["2", "-6"], "B": ["1", "7"], "C": ["4", "0"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(58)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + sqrt(10) + sqrt(58)/2)*sqrt(-sqrt(58)/2 + sqrt(10) + sqrt(170)/2)/sqrt(sqrt(10) + sqrt(58)/2 + sqrt(170)/2)", "incircle_radius_approx": 1.186119441537787, "source_dataset": "advanced_geometry", "source_index": 279, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(3, -6), B=(-8, -3), and C=(-1, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "38.45°", "metadata": {"A": ["3", "-6"], "B": ["-8", "-3"], "C": ["-1", "0"], "angle_ABC_degrees": 38.45370921670597, "source_dataset": "advanced_geometry", "source_index": 280, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -6), B=(9, -1), and C=(-9, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "41.32°", "metadata": {"A": ["-7", "-6"], "B": ["9", "-1"], "C": ["-9", "7"], "angle_ABC_degrees": 41.3165136108395, "source_dataset": "advanced_geometry", "source_index": 281, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, 5), B=(-8, 7), and C=(3, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.22°", "metadata": {"A": ["-5", "5"], "B": ["-8", "7"], "C": ["3", "0"], "angle_ABC_degrees": 1.218875235131298, "source_dataset": "advanced_geometry", "source_index": 282, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 8), B=(-7, -8), and C=(-5, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.95°", "metadata": {"A": ["-6", "8"], "B": ["-7", "-8"], "C": ["-5", "1"], "angle_ABC_degrees": 8.95247333415416, "source_dataset": "advanced_geometry", "source_index": 283, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, -1), B=(-9, 4), and C=(2, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.257, -1.681)", "metadata": {"A": ["5", "-1"], "B": ["-9", "4"], "C": ["2", "-8"], "ortho": ["481/113", "-190/113"], "orthocenter_approx": [4.256637168141593, -1.6814159292035398], "source_dataset": "advanced_geometry", "source_index": 284, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 4), B=(0, -6), and C=(-6, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.778, -8.667)", "metadata": {"A": ["6", "4"], "B": ["0", "-6"], "C": ["-6", "-4"], "ortho": ["16/9", "-26/3"], "orthocenter_approx": [1.7777777777777777, -8.666666666666666], "source_dataset": "advanced_geometry", "source_index": 285, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 0), B=(4, 5), and C=(-8, 5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "111.80°", "metadata": {"A": ["6", "0"], "B": ["4", "5"], "C": ["-8", "5"], "angle_ABC_degrees": 111.80140948635182, "source_dataset": "advanced_geometry", "source_index": 286, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, 6), B=(0, -3), and C=(2, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.958", "metadata": {"A": ["-1", "6"], "B": ["0", "-3"], "C": ["2", "-2"], "incircle_radius_exact": "sqrt(-sqrt(5)/2 + sqrt(73)/2 + sqrt(82)/2)*sqrt(-sqrt(73)/2 + sqrt(5)/2 + sqrt(82)/2)*sqrt(-sqrt(82)/2 + sqrt(5)/2 + sqrt(73)/2)/sqrt(sqrt(5)/2 + sqrt(73)/2 + sqrt(82)/2)", "incircle_radius_approx": 0.9578806343200847, "source_dataset": "advanced_geometry", "source_index": 287, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-8, -9), B=(2, 7), and C=(-10, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "27.74°", "metadata": {"A": ["-8", "-9"], "B": ["2", "7"], "C": ["-10", "0"], "angle_ABC_degrees": 27.73817962838724, "source_dataset": "advanced_geometry", "source_index": 288, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(0, -5), B=(-3, 10), and C=(-7, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "101.31°", "metadata": {"A": ["0", "-5"], "B": ["-3", "10"], "C": ["-7", "10"], "angle_ABC_degrees": 101.30993247402021, "source_dataset": "advanced_geometry", "source_index": 289, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, 6), B=(8, 3), and C=(8, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(14.000, 6.000)", "metadata": {"A": ["6", "6"], "B": ["8", "3"], "C": ["8", "2"], "ortho": ["14", "6"], "orthocenter_approx": [14.0, 6.0], "source_dataset": "advanced_geometry", "source_index": 290, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 1), B=(-4, 2), and C=(-8, -1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "131.82°", "metadata": {"A": ["1", "1"], "B": ["-4", "2"], "C": ["-8", "-1"], "angle_ABC_degrees": 131.82016988013578, "source_dataset": "advanced_geometry", "source_index": 291, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 2), B=(3, -6), and C=(1, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.909, 0.545)", "metadata": {"A": ["7", "2"], "B": ["3", "-6"], "C": ["1", "1"], "ortho": ["21/11", "6/11"], "orthocenter_approx": [1.9090909090909092, 0.5454545454545454], "source_dataset": "advanced_geometry", "source_index": 292, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 4), B=(7, 3), and C=(-9, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.252, 5.709)", "metadata": {"A": ["-10", "4"], "B": ["7", "3"], "C": ["-9", "10"], "ortho": ["-953/103", "588/103"], "orthocenter_approx": [-9.25242718446602, 5.70873786407767], "source_dataset": "advanced_geometry", "source_index": 293, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, -4), B=(-6, 7), and C=(1, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.177, -9.937)", "metadata": {"A": ["7", "-4"], "B": ["-6", "7"], "C": ["1", "-5"], "ortho": ["-251/79", "-785/79"], "orthocenter_approx": [-3.1772151898734178, -9.936708860759493], "source_dataset": "advanced_geometry", "source_index": 294, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 6), B=(1, -10), and C=(6, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, 7.250)", "metadata": {"A": ["5", "6"], "B": ["1", "-10"], "C": ["6", "6"], "ortho": ["1", "29/4"], "orthocenter_approx": [1.0, 7.25], "source_dataset": "advanced_geometry", "source_index": 295, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, -2), B=(4, 5), and C=(-4, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.667, 7.333)", "metadata": {"A": ["3", "-2"], "B": ["4", "5"], "C": ["-4", "9"], "ortho": ["23/3", "22/3"], "orthocenter_approx": [7.666666666666667, 7.333333333333333], "source_dataset": "advanced_geometry", "source_index": 296, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -6), B=(4, -9), and C=(6, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(14.222, -6.444)", "metadata": {"A": ["10", "-6"], "B": ["4", "-9"], "C": ["6", "10"], "ortho": ["128/9", "-58/9"], "orthocenter_approx": [14.222222222222221, -6.444444444444445], "source_dataset": "advanced_geometry", "source_index": 297, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, -7), B=(6, 7), and C=(-7, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.719", "metadata": {"A": ["8", "-7"], "B": ["6", "7"], "C": ["-7", "8"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + 25*sqrt(2)/2)*sqrt(-5*sqrt(2)/2 + sqrt(170)/2)*sqrt(5*sqrt(2)/2 + sqrt(170)/2)/sqrt(sqrt(170)/2 + 25*sqrt(2)/2)", "incircle_radius_approx": 3.719489041487013, "source_dataset": "advanced_geometry", "source_index": 298, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 8), B=(-6, -9), and C=(-9, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "27.01°", "metadata": {"A": ["-1", "8"], "B": ["-6", "-9"], "C": ["-9", "7"], "angle_ABC_degrees": 27.009195610189916, "source_dataset": "advanced_geometry", "source_index": 299, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 8), B=(-8, -8), and C=(-8, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-20.000, 8.000)", "metadata": {"A": ["-4", "8"], "B": ["-8", "-8"], "C": ["-8", "5"], "ortho": ["-20", "8"], "orthocenter_approx": [-20.0, 8.0], "source_dataset": "advanced_geometry", "source_index": 300, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, -5), B=(4, 8), and C=(-8, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.025, 6.848)", "metadata": {"A": ["5", "-5"], "B": ["4", "8"], "C": ["-8", "6"], "ortho": ["239/79", "541/79"], "orthocenter_approx": [3.0253164556962027, 6.848101265822785], "source_dataset": "advanced_geometry", "source_index": 301, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, -7), B=(-1, 6), and C=(9, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.209", "metadata": {"A": ["2", "-7"], "B": ["-1", "6"], "C": ["9", "0"], "incircle_radius_exact": "sqrt(-sqrt(178)/2 + 7*sqrt(2)/2 + sqrt(34))*sqrt(-7*sqrt(2)/2 + sqrt(34) + sqrt(178)/2)*sqrt(-sqrt(34) + 7*sqrt(2)/2 + sqrt(178)/2)/sqrt(7*sqrt(2)/2 + sqrt(34) + sqrt(178)/2)", "incircle_radius_approx": 3.2088874455657668, "source_dataset": "advanced_geometry", "source_index": 302, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -5), B=(3, -7), and C=(-4, 1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.789", "metadata": {"A": ["-10", "-5"], "B": ["3", "-7"], "C": ["-4", "1"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + 3*sqrt(2) + sqrt(173)/2)*sqrt(-sqrt(173)/2 + 3*sqrt(2) + sqrt(113)/2)*sqrt(-3*sqrt(2) + sqrt(113)/2 + sqrt(173)/2)/sqrt(3*sqrt(2) + sqrt(113)/2 + sqrt(173)/2)", "incircle_radius_approx": 2.789108650038751, "source_dataset": "advanced_geometry", "source_index": 303, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 9), B=(6, -7), and C=(7, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "40.14°", "metadata": {"A": ["-4", "9"], "B": ["6", "-7"], "C": ["7", "0"], "angle_ABC_degrees": 40.135485562239474, "source_dataset": "advanced_geometry", "source_index": 304, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 5), B=(-2, 5), and C=(-4, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.550", "metadata": {"A": ["9", "5"], "B": ["-2", "5"], "C": ["-4", "-3"], "incircle_radius_exact": "sqrt(-11/2 + sqrt(17) + sqrt(233)/2)*sqrt(-sqrt(17) + 11/2 + sqrt(233)/2)*sqrt(-sqrt(233)/2 + sqrt(17) + 11/2)/sqrt(sqrt(17) + 11/2 + sqrt(233)/2)", "incircle_radius_approx": 2.549944962539698, "source_dataset": "advanced_geometry", "source_index": 305, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, -4), B=(-3, 4), and C=(1, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.992", "metadata": {"A": ["-5", "-4"], "B": ["-3", "4"], "C": ["1", "-8"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + sqrt(13) + sqrt(17))*sqrt(-sqrt(13) + sqrt(17) + 2*sqrt(10))*sqrt(-sqrt(17) + sqrt(13) + 2*sqrt(10))/sqrt(sqrt(13) + sqrt(17) + 2*sqrt(10))", "incircle_radius_approx": 1.9924270379497566, "source_dataset": "advanced_geometry", "source_index": 306, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 3), B=(-1, 2), and C=(8, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.659", "metadata": {"A": ["6", "3"], "B": ["-1", "2"], "C": ["8", "-2"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(29)/2 + sqrt(97)/2)*sqrt(-sqrt(29)/2 + 5*sqrt(2)/2 + sqrt(97)/2)*sqrt(-sqrt(97)/2 + sqrt(29)/2 + 5*sqrt(2)/2)/sqrt(sqrt(29)/2 + 5*sqrt(2)/2 + sqrt(97)/2)", "incircle_radius_approx": 1.658814167617234, "source_dataset": "advanced_geometry", "source_index": 307, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 2), B=(0, 10), and C=(1, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.370", "metadata": {"A": ["-5", "2"], "B": ["0", "10"], "C": ["1", "-2"], "incircle_radius_exact": "sqrt(-sqrt(13) + sqrt(89)/2 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + sqrt(13) + sqrt(89)/2)*sqrt(-sqrt(89)/2 + sqrt(13) + sqrt(145)/2)/sqrt(sqrt(13) + sqrt(89)/2 + sqrt(145)/2)", "incircle_radius_approx": 2.370438270317465, "source_dataset": "advanced_geometry", "source_index": 308, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, 1), B=(0, -2), and C=(0, -1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "69.44°", "metadata": {"A": ["8", "1"], "B": ["0", "-2"], "C": ["0", "-1"], "angle_ABC_degrees": 69.44395478041653, "source_dataset": "advanced_geometry", "source_index": 309, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -8), B=(-2, -6), and C=(8, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.411", "metadata": {"A": ["-6", "-8"], "B": ["-2", "-6"], "C": ["8", "-4"], "incircle_radius_exact": "sqrt(-sqrt(26) + sqrt(5) + sqrt(53))*sqrt(-sqrt(5) + sqrt(26) + sqrt(53))*sqrt(-sqrt(53) + sqrt(5) + sqrt(26))/sqrt(sqrt(5) + sqrt(26) + sqrt(53))", "incircle_radius_approx": 0.41053157503418086, "source_dataset": "advanced_geometry", "source_index": 310, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 2), B=(-8, 9), and C=(5, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.000, 2.000)", "metadata": {"A": ["-8", "2"], "B": ["-8", "9"], "C": ["5", "2"], "ortho": ["-8", "2"], "orthocenter_approx": [-8.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 311, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, -8), B=(-7, 10), and C=(-6, -10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(10.286, -7.286)", "metadata": {"A": ["-4", "-8"], "B": ["-7", "10"], "C": ["-6", "-10"], "ortho": ["72/7", "-51/7"], "orthocenter_approx": [10.285714285714286, -7.285714285714286], "source_dataset": "advanced_geometry", "source_index": 312, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, 9), B=(2, 0), and C=(-9, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.412, 0.588)", "metadata": {"A": ["-7", "9"], "B": ["2", "0"], "C": ["-9", "-6"], "ortho": ["-41/17", "10/17"], "orthocenter_approx": [-2.411764705882353, 0.5882352941176471], "source_dataset": "advanced_geometry", "source_index": 313, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, 5), B=(-2, 3), and C=(3, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "113.86°", "metadata": {"A": ["-9", "5"], "B": ["-2", "3"], "C": ["3", "9"], "angle_ABC_degrees": 113.86017519134235, "source_dataset": "advanced_geometry", "source_index": 314, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, -8), B=(-5, -3), and C=(-2, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-27.000, -36.000)", "metadata": {"A": ["1", "-8"], "B": ["-5", "-3"], "C": ["-2", "-6"], "ortho": ["-27", "-36"], "orthocenter_approx": [-27.0, -36.0], "source_dataset": "advanced_geometry", "source_index": 315, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 2), B=(-1, -4), and C=(-10, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.286, -3.143)", "metadata": {"A": ["-4", "2"], "B": ["-1", "-4"], "C": ["-10", "-7"], "ortho": ["-16/7", "-22/7"], "orthocenter_approx": [-2.2857142857142856, -3.142857142857143], "source_dataset": "advanced_geometry", "source_index": 316, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 2), B=(8, 5), and C=(8, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.500, 2.000)", "metadata": {"A": ["6", "2"], "B": ["8", "5"], "C": ["8", "-1"], "ortho": ["7/2", "2"], "orthocenter_approx": [3.5, 2.0], "source_dataset": "advanced_geometry", "source_index": 317, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -3), B=(-8, -4), and C=(-6, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.667, -2.333)", "metadata": {"A": ["-1", "-3"], "B": ["-8", "-4"], "C": ["-6", "7"], "ortho": ["-14/3", "-7/3"], "orthocenter_approx": [-4.666666666666667, -2.3333333333333335], "source_dataset": "advanced_geometry", "source_index": 318, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 2), B=(2, 4), and C=(-10, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "9.27°", "metadata": {"A": ["-6", "2"], "B": ["2", "4"], "C": ["-10", "3"], "angle_ABC_degrees": 9.2726017772003, "source_dataset": "advanced_geometry", "source_index": 319, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 9), B=(1, -3), and C=(6, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.58°", "metadata": {"A": ["6", "9"], "B": ["1", "-3"], "C": ["6", "10"], "angle_ABC_degrees": 1.5823539226186096, "source_dataset": "advanced_geometry", "source_index": 320, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, 7), B=(-5, -1), and C=(-8, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.692, -0.154)", "metadata": {"A": ["-9", "7"], "B": ["-5", "-1"], "C": ["-8", "-8"], "ortho": ["100/13", "-2/13"], "orthocenter_approx": [7.6923076923076925, -0.15384615384615385], "source_dataset": "advanced_geometry", "source_index": 321, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 5), B=(-10, -10), and C=(-5, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.545, -1.273)", "metadata": {"A": ["-8", "5"], "B": ["-10", "-10"], "C": ["-5", "0"], "ortho": ["50/11", "-14/11"], "orthocenter_approx": [4.545454545454545, -1.2727272727272727], "source_dataset": "advanced_geometry", "source_index": 322, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -6), B=(-1, 9), and C=(-4, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["-1", "-6"], "B": ["-1", "9"], "C": ["-4", "6"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 323, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, -7), B=(-6, 6), and C=(2, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(15.000, 13.000)", "metadata": {"A": ["5", "-7"], "B": ["-6", "6"], "C": ["2", "2"], "ortho": ["15", "13"], "orthocenter_approx": [15.0, 13.0], "source_dataset": "advanced_geometry", "source_index": 324, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 5), B=(-10, 1), and C=(-3, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.000, -2.000)", "metadata": {"A": ["6", "5"], "B": ["-10", "1"], "C": ["-3", "-10"], "ortho": ["-5", "-2"], "orthocenter_approx": [-5.0, -2.0], "source_dataset": "advanced_geometry", "source_index": 325, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, -9), B=(-9, -4), and C=(-7, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.435", "metadata": {"A": ["8", "-9"], "B": ["-9", "-4"], "C": ["-7", "5"], "incircle_radius_exact": "sqrt(-sqrt(314)/2 + sqrt(85)/2 + sqrt(421)/2)*sqrt(-sqrt(421)/2 + sqrt(85)/2 + sqrt(314)/2)*sqrt(-sqrt(85)/2 + sqrt(314)/2 + sqrt(421)/2)/sqrt(sqrt(85)/2 + sqrt(314)/2 + sqrt(421)/2)", "incircle_radius_approx": 3.4346249801331745, "source_dataset": "advanced_geometry", "source_index": 326, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, -3), B=(9, -9), and C=(-10, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.133", "metadata": {"A": ["-1", "-3"], "B": ["9", "-9"], "C": ["-10", "3"], "incircle_radius_exact": "sqrt(-3*sqrt(13)/2 + sqrt(34) + sqrt(505)/2)*sqrt(-sqrt(34) + 3*sqrt(13)/2 + sqrt(505)/2)*sqrt(-sqrt(505)/2 + 3*sqrt(13)/2 + sqrt(34))/sqrt(3*sqrt(13)/2 + sqrt(34) + sqrt(505)/2)", "incircle_radius_approx": 0.13347938151805286, "source_dataset": "advanced_geometry", "source_index": 327, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, -1), B=(8, 6), and C=(-2, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(13.571, -3.551)", "metadata": {"A": ["10", "-1"], "B": ["8", "6"], "C": ["-2", "-8"], "ortho": ["95/7", "-174/49"], "orthocenter_approx": [13.571428571428571, -3.5510204081632653], "source_dataset": "advanced_geometry", "source_index": 328, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 3), B=(3, 9), and C=(-7, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.204", "metadata": {"A": ["9", "3"], "B": ["3", "9"], "C": ["-7", "-3"], "incircle_radius_exact": "sqrt(-3*sqrt(2) + sqrt(61) + sqrt(73))*sqrt(-sqrt(61) + 3*sqrt(2) + sqrt(73))*sqrt(-sqrt(73) + 3*sqrt(2) + sqrt(61))/sqrt(3*sqrt(2) + sqrt(61) + sqrt(73))", "incircle_radius_approx": 3.204366622114373, "source_dataset": "advanced_geometry", "source_index": 329, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, -3), B=(-8, 6), and C=(-4, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(13.000, 21.000)", "metadata": {"A": ["1", "-3"], "B": ["-8", "6"], "C": ["-4", "4"], "ortho": ["13", "21"], "orthocenter_approx": [13.0, 21.0], "source_dataset": "advanced_geometry", "source_index": 330, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, -7), B=(-4, -3), and C=(9, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-91.000, 84.000)", "metadata": {"A": ["-7", "-7"], "B": ["-4", "-3"], "C": ["9", "9"], "ortho": ["-91", "84"], "orthocenter_approx": [-91.0, 84.0], "source_dataset": "advanced_geometry", "source_index": 331, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, 6), B=(-6, -6), and C=(6, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.450", "metadata": {"A": ["-6", "6"], "B": ["-6", "-6"], "C": ["6", "7"], "incircle_radius_exact": "sqrt(-6 + sqrt(145)/2 + sqrt(313)/2)*sqrt(-sqrt(145)/2 + 6 + sqrt(313)/2)*sqrt(-sqrt(313)/2 + 6 + sqrt(145)/2)/sqrt(6 + sqrt(145)/2 + sqrt(313)/2)", "incircle_radius_approx": 3.450473672363012, "source_dataset": "advanced_geometry", "source_index": 332, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, -3), B=(3, -5), and C=(-2, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, -2.000)", "metadata": {"A": ["1", "-3"], "B": ["3", "-5"], "C": ["-2", "-5"], "ortho": ["1", "-2"], "orthocenter_approx": [1.0, -2.0], "source_dataset": "advanced_geometry", "source_index": 333, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, 5), B=(-1, -5), and C=(6, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "89.25°", "metadata": {"A": ["2", "5"], "B": ["-1", "-5"], "C": ["6", "-7"], "angle_ABC_degrees": 89.24615166692922, "source_dataset": "advanced_geometry", "source_index": 334, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, -7), B=(-3, -8), and C=(-10, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.839, -14.452)", "metadata": {"A": ["6", "-7"], "B": ["-3", "-8"], "C": ["-10", "5"], "ortho": ["-243/31", "-448/31"], "orthocenter_approx": [-7.838709677419355, -14.451612903225806], "source_dataset": "advanced_geometry", "source_index": 335, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, 2), B=(4, -6), and C=(4, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.400, 2.000)", "metadata": {"A": ["-1", "2"], "B": ["4", "-6"], "C": ["4", "3"], "ortho": ["12/5", "2"], "orthocenter_approx": [2.4, 2.0], "source_dataset": "advanced_geometry", "source_index": 336, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 7), B=(6, 0), and C=(8, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.000, -17.143)", "metadata": {"A": ["-4", "7"], "B": ["6", "0"], "C": ["8", "0"], "ortho": ["-4", "-120/7"], "orthocenter_approx": [-4.0, -17.142857142857142], "source_dataset": "advanced_geometry", "source_index": 337, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 8), B=(3, 7), and C=(4, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.931, 8.793)", "metadata": {"A": ["0", "8"], "B": ["3", "7"], "C": ["4", "-3"], "ortho": ["230/29", "255/29"], "orthocenter_approx": [7.931034482758621, 8.793103448275861], "source_dataset": "advanced_geometry", "source_index": 338, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 8), B=(-10, -4), and C=(-7, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.234", "metadata": {"A": ["5", "8"], "B": ["-10", "-4"], "C": ["-7", "-1"], "incircle_radius_exact": "sqrt(-15/2 + 3*sqrt(2)/2 + 3*sqrt(41)/2)*sqrt(-3*sqrt(2)/2 + 15/2 + 3*sqrt(41)/2)*sqrt(-3*sqrt(41)/2 + 3*sqrt(2)/2 + 15/2)/sqrt(3*sqrt(2)/2 + 15/2 + 3*sqrt(41)/2)", "incircle_radius_approx": 0.23405796483302643, "source_dataset": "advanced_geometry", "source_index": 339, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -6), B=(0, -3), and C=(6, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.600, -4.600)", "metadata": {"A": ["-2", "-6"], "B": ["0", "-3"], "C": ["6", "-9"], "ortho": ["-3/5", "-23/5"], "orthocenter_approx": [-0.6, -4.6], "source_dataset": "advanced_geometry", "source_index": 340, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 2), B=(-6, -3), and C=(5, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "42.60°", "metadata": {"A": ["6", "2"], "B": ["-6", "-3"], "C": ["5", "-7"], "angle_ABC_degrees": 42.6029714699404, "source_dataset": "advanced_geometry", "source_index": 341, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -9), B=(7, -10), and C=(10, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "93.95°", "metadata": {"A": ["2", "-9"], "B": ["7", "-10"], "C": ["10", "1"], "angle_ABC_degrees": 93.94518622903756, "source_dataset": "advanced_geometry", "source_index": 342, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, 2), B=(6, -1), and C=(-10, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.000, 7.333)", "metadata": {"A": ["5", "2"], "B": ["6", "-1"], "C": ["-10", "2"], "ortho": ["6", "22/3"], "orthocenter_approx": [6.0, 7.333333333333333], "source_dataset": "advanced_geometry", "source_index": 343, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-1, 2), B=(7, 4), and C=(3, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "58.86°", "metadata": {"A": ["-1", "2"], "B": ["7", "4"], "C": ["3", "-9"], "angle_ABC_degrees": 58.861027563021146, "source_dataset": "advanced_geometry", "source_index": 344, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -9), B=(-5, -4), and C=(-4, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.495", "metadata": {"A": ["-4", "-9"], "B": ["-5", "-4"], "C": ["-4", "1"], "incircle_radius_exact": "sqrt(5)*sqrt(-5 + sqrt(26))*sqrt(25 + 5*sqrt(26))/(5 + sqrt(26))", "incircle_radius_approx": 0.4950975679639241, "source_dataset": "advanced_geometry", "source_index": 345, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 1), B=(-5, -8), and C=(4, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.953", "metadata": {"A": ["1", "1"], "B": ["-5", "-8"], "C": ["4", "-3"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(106)/2 + 3*sqrt(13)/2)*sqrt(-sqrt(106)/2 + 5/2 + 3*sqrt(13)/2)*sqrt(-3*sqrt(13)/2 + 5/2 + sqrt(106)/2)/sqrt(5/2 + sqrt(106)/2 + 3*sqrt(13)/2)", "incircle_radius_approx": 1.953103760043061, "source_dataset": "advanced_geometry", "source_index": 346, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, 1), B=(8, 3), and C=(7, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "54.46°", "metadata": {"A": ["10", "1"], "B": ["8", "3"], "C": ["7", "-3"], "angle_ABC_degrees": 54.46232220802562, "source_dataset": "advanced_geometry", "source_index": 347, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, -8), B=(6, 7), and C=(9, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "63.97°", "metadata": {"A": ["8", "-8"], "B": ["6", "7"], "C": ["9", "6"], "angle_ABC_degrees": 63.970407808486556, "source_dataset": "advanced_geometry", "source_index": 348, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 10), B=(-3, -6), and C=(-5, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.310", "metadata": {"A": ["-5", "10"], "B": ["-3", "-6"], "C": ["-5", "5"], "incircle_radius_exact": "sqrt(-5/2 + 5*sqrt(5)/2 + sqrt(65))*sqrt(-5*sqrt(5)/2 + 5/2 + sqrt(65))*sqrt(-sqrt(65) + 5/2 + 5*sqrt(5)/2)/sqrt(5/2 + 5*sqrt(5)/2 + sqrt(65))", "incircle_radius_approx": 0.30955099105390466, "source_dataset": "advanced_geometry", "source_index": 349, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, 4), B=(9, -4), and C=(5, -4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "57.99°", "metadata": {"A": ["4", "4"], "B": ["9", "-4"], "C": ["5", "-4"], "angle_ABC_degrees": 57.9946167919165, "source_dataset": "advanced_geometry", "source_index": 350, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, -5), B=(-10, 3), and C=(4, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.925", "metadata": {"A": ["0", "-5"], "B": ["-10", "3"], "C": ["4", "8"], "incircle_radius_exact": "sqrt(-sqrt(185)/2 + sqrt(41) + sqrt(221)/2)*sqrt(-sqrt(221)/2 + sqrt(41) + sqrt(185)/2)*sqrt(-sqrt(41) + sqrt(185)/2 + sqrt(221)/2)/sqrt(sqrt(41) + sqrt(185)/2 + sqrt(221)/2)", "incircle_radius_approx": 3.9250092832802963, "source_dataset": "advanced_geometry", "source_index": 351, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-3, 1), B=(-4, 5), and C=(-8, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.238, 1.190)", "metadata": {"A": ["-3", "1"], "B": ["-4", "5"], "C": ["-8", "0"], "ortho": ["-68/21", "25/21"], "orthocenter_approx": [-3.238095238095238, 1.1904761904761905], "source_dataset": "advanced_geometry", "source_index": 352, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -10), B=(-6, 9), and C=(9, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.681", "metadata": {"A": ["0", "-10"], "B": ["-6", "9"], "C": ["9", "-7"], "incircle_radius_exact": "sqrt(-3*sqrt(10)/2 + sqrt(397)/2 + sqrt(481)/2)*sqrt(-sqrt(397)/2 + 3*sqrt(10)/2 + sqrt(481)/2)*sqrt(-sqrt(481)/2 + 3*sqrt(10)/2 + sqrt(397)/2)/sqrt(3*sqrt(10)/2 + sqrt(397)/2 + sqrt(481)/2)", "incircle_radius_approx": 3.681096015906262, "source_dataset": "advanced_geometry", "source_index": 353, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, 10), B=(-2, 7), and C=(-6, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(11.500, 1.000)", "metadata": {"A": ["-2", "10"], "B": ["-2", "7"], "C": ["-6", "1"], "ortho": ["23/2", "1"], "orthocenter_approx": [11.5, 1.0], "source_dataset": "advanced_geometry", "source_index": 354, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 2), B=(4, -10), and C=(-2, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.000, 2.000)", "metadata": {"A": ["-8", "2"], "B": ["4", "-10"], "C": ["-2", "8"], "ortho": ["-8", "2"], "orthocenter_approx": [-8.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 355, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -3), B=(9, 4), and C=(1, 4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.286", "metadata": {"A": ["3", "-3"], "B": ["9", "4"], "C": ["1", "4"], "incircle_radius_exact": "sqrt(-4 + sqrt(53)/2 + sqrt(85)/2)*sqrt(-sqrt(53)/2 + 4 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + sqrt(53)/2 + 4)/sqrt(sqrt(53)/2 + 4 + sqrt(85)/2)", "incircle_radius_approx": 2.2857465337192533, "source_dataset": "advanced_geometry", "source_index": 356, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 9), B=(-10, 9), and C=(-3, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, 9.875)", "metadata": {"A": ["-5", "9"], "B": ["-10", "9"], "C": ["-3", "-7"], "ortho": ["-3", "79/8"], "orthocenter_approx": [-3.0, 9.875], "source_dataset": "advanced_geometry", "source_index": 357, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -2), B=(-8, 8), and C=(5, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.844", "metadata": {"A": ["7", "-2"], "B": ["-8", "8"], "C": ["5", "4"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(185)/2 + 5*sqrt(13)/2)*sqrt(-5*sqrt(13)/2 + sqrt(10) + sqrt(185)/2)*sqrt(-sqrt(185)/2 + sqrt(10) + 5*sqrt(13)/2)/sqrt(sqrt(10) + sqrt(185)/2 + 5*sqrt(13)/2)", "incircle_radius_approx": 1.844348466230347, "source_dataset": "advanced_geometry", "source_index": 358, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 3), B=(-2, 8), and C=(2, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.644", "metadata": {"A": ["-7", "3"], "B": ["-2", "8"], "C": ["2", "5"], "incircle_radius_exact": "sqrt(-5/2 + 5*sqrt(2)/2 + sqrt(85)/2)*sqrt(-5*sqrt(2)/2 + 5/2 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + 5/2 + 5*sqrt(2)/2)/sqrt(5/2 + 5*sqrt(2)/2 + sqrt(85)/2)", "incircle_radius_approx": 1.6439170258481055, "source_dataset": "advanced_geometry", "source_index": 359, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 10), B=(3, 2), and C=(-1, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.695", "metadata": {"A": ["-7", "10"], "B": ["3", "2"], "C": ["-1", "7"], "incircle_radius_exact": "sqrt(-sqrt(41)/2 + 3*sqrt(5)/2)*sqrt(sqrt(41)/2 + 3*sqrt(5)/2)*sqrt(-3*sqrt(5)/2 + 3*sqrt(41)/2)/sqrt(3*sqrt(5)/2 + 3*sqrt(41)/2)", "incircle_radius_approx": 0.6945093766555098, "source_dataset": "advanced_geometry", "source_index": 360, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, 0), B=(-9, -5), and C=(-2, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.931, 8.352)", "metadata": {"A": ["9", "0"], "B": ["-9", "-5"], "C": ["-2", "5"], "ortho": ["-85/29", "1211/145"], "orthocenter_approx": [-2.9310344827586206, 8.351724137931035], "source_dataset": "advanced_geometry", "source_index": 361, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, -10), B=(6, -8), and C=(9, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-15.000, 41.000)", "metadata": {"A": ["2", "-10"], "B": ["6", "-8"], "C": ["9", "-7"], "ortho": ["-15", "41"], "orthocenter_approx": [-15.0, 41.0], "source_dataset": "advanced_geometry", "source_index": 362, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 4), B=(-7, 9), and C=(8, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.310, 1.676)", "metadata": {"A": ["-5", "4"], "B": ["-7", "9"], "C": ["8", "7"], "ortho": ["-377/71", "119/71"], "orthocenter_approx": [-5.309859154929577, 1.676056338028169], "source_dataset": "advanced_geometry", "source_index": 363, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, 2), B=(8, -1), and C=(10, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.019", "metadata": {"A": ["3", "2"], "B": ["8", "-1"], "C": ["10", "-7"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(34)/2 + sqrt(130)/2)*sqrt(-sqrt(130)/2 + sqrt(34)/2 + sqrt(10))*sqrt(-sqrt(34)/2 + sqrt(10) + sqrt(130)/2)/sqrt(sqrt(34)/2 + sqrt(10) + sqrt(130)/2)", "incircle_radius_approx": 1.0187941427089182, "source_dataset": "advanced_geometry", "source_index": 364, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -1), B=(7, 2), and C=(1, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "61.49°", "metadata": {"A": ["-7", "-1"], "B": ["7", "2"], "C": ["1", "9"], "angle_ABC_degrees": 61.49346243200763, "source_dataset": "advanced_geometry", "source_index": 365, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, -2), B=(7, -3), and C=(8, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "76.76°", "metadata": {"A": ["4", "-2"], "B": ["7", "-3"], "C": ["8", "8"], "angle_ABC_degrees": 76.7594800848128, "source_dataset": "advanced_geometry", "source_index": 366, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(7, 5), B=(-6, 7), and C=(3, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "27.18°", "metadata": {"A": ["7", "5"], "B": ["-6", "7"], "C": ["3", "10"], "angle_ABC_degrees": 27.18111108547722, "source_dataset": "advanced_geometry", "source_index": 367, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -10), B=(0, -8), and C=(-1, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(23.000, -8.000)", "metadata": {"A": ["-1", "-10"], "B": ["0", "-8"], "C": ["-1", "4"], "ortho": ["23", "-8"], "orthocenter_approx": [23.0, -8.0], "source_dataset": "advanced_geometry", "source_index": 368, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, 0), B=(-4, 9), and C=(-4, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.500, 0.000)", "metadata": {"A": ["-10", "0"], "B": ["-4", "9"], "C": ["-4", "-3"], "ortho": ["-17/2", "0"], "orthocenter_approx": [-8.5, 0.0], "source_dataset": "advanced_geometry", "source_index": 369, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, -7), B=(-4, -7), and C=(-1, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.843", "metadata": {"A": ["-9", "-7"], "B": ["-4", "-7"], "C": ["-1", "-4"], "incircle_radius_exact": "sqrt(-5/2 + 3*sqrt(2)/2 + sqrt(73)/2)*sqrt(-3*sqrt(2)/2 + 5/2 + sqrt(73)/2)*sqrt(-sqrt(73)/2 + 3*sqrt(2)/2 + 5/2)/sqrt(3*sqrt(2)/2 + 5/2 + sqrt(73)/2)", "incircle_radius_approx": 0.8433293900363286, "source_dataset": "advanced_geometry", "source_index": 370, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, -5), B=(-8, 7), and C=(-1, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "14.74°", "metadata": {"A": ["4", "-5"], "B": ["-8", "7"], "C": ["-1", "-5"], "angle_ABC_degrees": 14.743562836470735, "source_dataset": "advanced_geometry", "source_index": 371, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 9), B=(-3, 9), and C=(-2, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.633", "metadata": {"A": ["1", "9"], "B": ["-3", "9"], "C": ["-2", "-1"], "incircle_radius_exact": "sqrt(-2 + sqrt(101)/2 + sqrt(109)/2)*sqrt(-sqrt(101)/2 + 2 + sqrt(109)/2)*sqrt(-sqrt(109)/2 + 2 + sqrt(101)/2)/sqrt(2 + sqrt(101)/2 + sqrt(109)/2)", "incircle_radius_approx": 1.6333075755671667, "source_dataset": "advanced_geometry", "source_index": 372, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, -2), B=(-4, -4), and C=(7, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.992", "metadata": {"A": ["-3", "-2"], "B": ["-4", "-4"], "C": ["7", "-6"], "incircle_radius_exact": "sqrt(-sqrt(29) + 3*sqrt(5))*sqrt(-2*sqrt(5) + sqrt(29))*sqrt(2*sqrt(5) + sqrt(29))/sqrt(sqrt(29) + 3*sqrt(5))", "incircle_radius_approx": 0.9922793440236488, "source_dataset": "advanced_geometry", "source_index": 373, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 6), B=(8, 9), and C=(8, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.785", "metadata": {"A": ["-10", "6"], "B": ["8", "9"], "C": ["8", "-1"], "incircle_radius_exact": "sqrt(-5 + 3*sqrt(37)/2 + sqrt(373)/2)*sqrt(-3*sqrt(37)/2 + 5 + sqrt(373)/2)*sqrt(-sqrt(373)/2 + 5 + 3*sqrt(37)/2)/sqrt(5 + 3*sqrt(37)/2 + sqrt(373)/2)", "incircle_radius_approx": 3.7845740148048486, "source_dataset": "advanced_geometry", "source_index": 374, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, -9), B=(2, -1), and C=(-9, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.96°", "metadata": {"A": ["-8", "-9"], "B": ["2", "-1"], "C": ["-9", "1"], "angle_ABC_degrees": 48.96465472285612, "source_dataset": "advanced_geometry", "source_index": 375, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, -7), B=(-1, -6), and C=(-2, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.833", "metadata": {"A": ["1", "-7"], "B": ["-1", "-6"], "C": ["-2", "6"], "incircle_radius_exact": "sqrt(-sqrt(145)/2 + sqrt(5)/2 + sqrt(178)/2)*sqrt(-sqrt(178)/2 + sqrt(5)/2 + sqrt(145)/2)*sqrt(-sqrt(5)/2 + sqrt(145)/2 + sqrt(178)/2)/sqrt(sqrt(5)/2 + sqrt(145)/2 + sqrt(178)/2)", "incircle_radius_approx": 0.8327502084354423, "source_dataset": "advanced_geometry", "source_index": 376, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -6), B=(-1, 1), and C=(6, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "73.41°", "metadata": {"A": ["-6", "-6"], "B": ["-1", "1"], "C": ["6", "-8"], "angle_ABC_degrees": 73.41266144307258, "source_dataset": "advanced_geometry", "source_index": 377, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, -2), B=(2, -7), and C=(-7, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.000, -8.000)", "metadata": {"A": ["1", "-2"], "B": ["2", "-7"], "C": ["-7", "-10"], "ortho": ["3", "-8"], "orthocenter_approx": [3.0, -8.0], "source_dataset": "advanced_geometry", "source_index": 378, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 9), B=(-2, 5), and C=(5, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "64.98°", "metadata": {"A": ["9", "9"], "B": ["-2", "5"], "C": ["5", "-2"], "angle_ABC_degrees": 64.98310652189998, "source_dataset": "advanced_geometry", "source_index": 379, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, -10), B=(0, 6), and C=(9, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.685", "metadata": {"A": ["-1", "-10"], "B": ["0", "6"], "C": ["9", "-8"], "incircle_radius_exact": "sqrt(-sqrt(257)/2 + sqrt(26) + sqrt(277)/2)*sqrt(-sqrt(26) + sqrt(257)/2 + sqrt(277)/2)*sqrt(-sqrt(277)/2 + sqrt(26) + sqrt(257)/2)/sqrt(sqrt(26) + sqrt(257)/2 + sqrt(277)/2)", "incircle_radius_approx": 3.68533959033751, "source_dataset": "advanced_geometry", "source_index": 380, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 7), B=(10, 8), and C=(6, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.142", "metadata": {"A": ["-5", "7"], "B": ["10", "8"], "C": ["6", "3"], "incircle_radius_exact": "sqrt(-sqrt(137)/2 + sqrt(41)/2 + sqrt(226)/2)*sqrt(-sqrt(226)/2 + sqrt(41)/2 + sqrt(137)/2)*sqrt(-sqrt(41)/2 + sqrt(137)/2 + sqrt(226)/2)/sqrt(sqrt(41)/2 + sqrt(137)/2 + sqrt(226)/2)", "incircle_radius_approx": 2.1423536341558895, "source_dataset": "advanced_geometry", "source_index": 381, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, -6), B=(-9, 10), and C=(3, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "5.40°", "metadata": {"A": ["4", "-6"], "B": ["-9", "10"], "C": ["3", "-8"], "angle_ABC_degrees": 5.403791360249714, "source_dataset": "advanced_geometry", "source_index": 382, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, -5), B=(1, -9), and C=(9, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.625", "metadata": {"A": ["-8", "-5"], "B": ["1", "-9"], "C": ["9", "7"], "incircle_radius_exact": "sqrt(-sqrt(433)/2 + sqrt(97)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(97)/2 + sqrt(433)/2)*sqrt(-sqrt(97)/2 + 4*sqrt(5) + sqrt(433)/2)/sqrt(sqrt(97)/2 + 4*sqrt(5) + sqrt(433)/2)", "incircle_radius_approx": 3.6254234216833185, "source_dataset": "advanced_geometry", "source_index": 383, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 1), B=(-8, -4), and C=(-7, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-10.421, 4.474)", "metadata": {"A": ["0", "1"], "B": ["-8", "-4"], "C": ["-7", "-1"], "ortho": ["-198/19", "85/19"], "orthocenter_approx": [-10.421052631578947, 4.473684210526316], "source_dataset": "advanced_geometry", "source_index": 384, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, 5), B=(1, -9), and C=(-5, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.595, -2.946)", "metadata": {"A": ["9", "5"], "B": ["1", "-9"], "C": ["-5", "-1"], "ortho": ["-59/37", "-109/37"], "orthocenter_approx": [-1.5945945945945945, -2.945945945945946], "source_dataset": "advanced_geometry", "source_index": 385, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 6), B=(4, -7), and C=(9, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.879", "metadata": {"A": ["-9", "6"], "B": ["4", "-7"], "C": ["9", "-2"], "incircle_radius_exact": "sqrt(-4*sqrt(2) + sqrt(97))*sqrt(4*sqrt(2) + sqrt(97))*sqrt(-sqrt(97) + 9*sqrt(2))/sqrt(sqrt(97) + 9*sqrt(2))", "incircle_radius_approx": 2.879064259561751, "source_dataset": "advanced_geometry", "source_index": 386, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -10), B=(0, -10), and C=(-3, 5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "78.69°", "metadata": {"A": ["-7", "-10"], "B": ["0", "-10"], "C": ["-3", "5"], "angle_ABC_degrees": 78.69006752597979, "source_dataset": "advanced_geometry", "source_index": 387, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -10), B=(-6, -4), and C=(4, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.941", "metadata": {"A": ["-6", "-10"], "B": ["-6", "-4"], "C": ["4", "0"], "incircle_radius_exact": "sqrt(-3 + sqrt(29) + 5*sqrt(2))*sqrt(-5*sqrt(2) + 3 + sqrt(29))*sqrt(-sqrt(29) + 3 + 5*sqrt(2))/sqrt(3 + sqrt(29) + 5*sqrt(2))", "incircle_radius_approx": 1.9409645765244057, "source_dataset": "advanced_geometry", "source_index": 388, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -9), B=(-10, 3), and C=(10, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.450", "metadata": {"A": ["-10", "-9"], "B": ["-10", "3"], "C": ["10", "-5"], "incircle_radius_exact": "sqrt(-6 + 2*sqrt(26) + 2*sqrt(29))*sqrt(-2*sqrt(26) + 6 + 2*sqrt(29))*sqrt(-2*sqrt(29) + 6 + 2*sqrt(26))/sqrt(6 + 2*sqrt(26) + 2*sqrt(29))", "incircle_radius_approx": 4.449657359531245, "source_dataset": "advanced_geometry", "source_index": 389, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, 5), B=(-8, 2), and C=(5, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "37.76°", "metadata": {"A": ["10", "5"], "B": ["-8", "2"], "C": ["5", "-5"], "angle_ABC_degrees": 37.763077974031994, "source_dataset": "advanced_geometry", "source_index": 390, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, -8), B=(-9, 5), and C=(9, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.371, -1.831)", "metadata": {"A": ["5", "-8"], "B": ["-9", "5"], "C": ["9", "1"], "ortho": ["567/89", "-163/89"], "orthocenter_approx": [6.370786516853933, -1.8314606741573034], "source_dataset": "advanced_geometry", "source_index": 391, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, 6), B=(-4, 2), and C=(6, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.882", "metadata": {"A": ["-8", "6"], "B": ["-4", "2"], "C": ["6", "-1"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + 2*sqrt(2) + 7*sqrt(5)/2)*sqrt(-2*sqrt(2) + sqrt(109)/2 + 7*sqrt(5)/2)*sqrt(-7*sqrt(5)/2 + 2*sqrt(2) + sqrt(109)/2)/sqrt(2*sqrt(2) + sqrt(109)/2 + 7*sqrt(5)/2)", "incircle_radius_approx": 0.8818998576886075, "source_dataset": "advanced_geometry", "source_index": 392, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 0), B=(7, 7), and C=(-6, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(24.180, -16.623)", "metadata": {"A": ["5", "0"], "B": ["7", "7"], "C": ["-6", "-8"], "ortho": ["1475/61", "-1014/61"], "orthocenter_approx": [24.18032786885246, -16.62295081967213], "source_dataset": "advanced_geometry", "source_index": 393, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 2), B=(-3, -5), and C=(1, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.558", "metadata": {"A": ["5", "2"], "B": ["-3", "-5"], "C": ["1", "-3"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + sqrt(5) + sqrt(41)/2)*sqrt(-sqrt(41)/2 + sqrt(5) + sqrt(113)/2)*sqrt(-sqrt(5) + sqrt(41)/2 + sqrt(113)/2)/sqrt(sqrt(5) + sqrt(41)/2 + sqrt(113)/2)", "incircle_radius_approx": 0.5579992303849913, "source_dataset": "advanced_geometry", "source_index": 394, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -7), B=(-10, -6), and C=(-10, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "96.34°", "metadata": {"A": ["-1", "-7"], "B": ["-10", "-6"], "C": ["-10", "8"], "angle_ABC_degrees": 96.3401917459099, "source_dataset": "advanced_geometry", "source_index": 395, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 8), B=(6, -8), and C=(-1, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.165, 6.677)", "metadata": {"A": ["7", "8"], "B": ["6", "-8"], "C": ["-1", "7"], "ortho": ["529/127", "848/127"], "orthocenter_approx": [4.165354330708661, 6.677165354330708], "source_dataset": "advanced_geometry", "source_index": 396, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -1), B=(3, -2), and C=(6, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.111", "metadata": {"A": ["0", "-1"], "B": ["3", "-2"], "C": ["6", "3"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(34)/2 + sqrt(13))*sqrt(-sqrt(13) + sqrt(10)/2 + sqrt(34)/2)*sqrt(-sqrt(34)/2 + sqrt(10)/2 + sqrt(13))/sqrt(sqrt(10)/2 + sqrt(34)/2 + sqrt(13))", "incircle_radius_approx": 1.1108140639378727, "source_dataset": "advanced_geometry", "source_index": 397, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, -1), B=(-1, -3), and C=(-4, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.385, -8.846)", "metadata": {"A": ["-8", "-1"], "B": ["-1", "-3"], "C": ["-4", "-4"], "ortho": ["-70/13", "-115/13"], "orthocenter_approx": [-5.384615384615385, -8.846153846153847], "source_dataset": "advanced_geometry", "source_index": 398, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, 9), B=(-6, 5), and C=(-4, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.250, 0.500)", "metadata": {"A": ["2", "9"], "B": ["-6", "5"], "C": ["-4", "4"], "ortho": ["-9/4", "1/2"], "orthocenter_approx": [-2.25, 0.5], "source_dataset": "advanced_geometry", "source_index": 399, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, 9), B=(4, 3), and C=(3, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.485", "metadata": {"A": ["4", "9"], "B": ["4", "3"], "C": ["3", "5"], "incircle_radius_exact": "sqrt(-3 + sqrt(5)/2 + sqrt(17)/2)*sqrt(-sqrt(17)/2 + sqrt(5)/2 + 3)*sqrt(-sqrt(5)/2 + sqrt(17)/2 + 3)/sqrt(sqrt(5)/2 + sqrt(17)/2 + 3)", "incircle_radius_approx": 0.48546935197079627, "source_dataset": "advanced_geometry", "source_index": 400, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 10), B=(10, -2), and C=(0, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-13.320, -23.200)", "metadata": {"A": ["-10", "10"], "B": ["10", "-2"], "C": ["0", "-1"], "ortho": ["-333/25", "-116/5"], "orthocenter_approx": [-13.32, -23.2], "source_dataset": "advanced_geometry", "source_index": 401, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -10), B=(-1, 5), and C=(-2, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.498", "metadata": {"A": ["-1", "-10"], "B": ["-1", "5"], "C": ["-2", "-1"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(37)/2 + sqrt(82)/2)*sqrt(-sqrt(37)/2 + sqrt(82)/2 + 15/2)*sqrt(-sqrt(82)/2 + sqrt(37)/2 + 15/2)/sqrt(sqrt(37)/2 + sqrt(82)/2 + 15/2)", "incircle_radius_approx": 0.4977080929134155, "source_dataset": "advanced_geometry", "source_index": 402, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, -3), B=(-8, 2), and C=(-2, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "24.44°", "metadata": {"A": ["3", "-3"], "B": ["-8", "2"], "C": ["-2", "2"], "angle_ABC_degrees": 24.443954780416536, "source_dataset": "advanced_geometry", "source_index": 403, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, 10), B=(9, 10), and C=(-4, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.742", "metadata": {"A": ["7", "10"], "B": ["9", "10"], "C": ["-4", "-5"], "incircle_radius_exact": "sqrt(-1 + sqrt(346)/2 + sqrt(394)/2)*sqrt(-sqrt(346)/2 + 1 + sqrt(394)/2)*sqrt(-sqrt(394)/2 + 1 + sqrt(346)/2)/sqrt(1 + sqrt(346)/2 + sqrt(394)/2)", "incircle_radius_approx": 0.7416470429676211, "source_dataset": "advanced_geometry", "source_index": 404, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, -10), B=(9, 2), and C=(3, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.254", "metadata": {"A": ["6", "-10"], "B": ["9", "2"], "C": ["3", "10"], "incircle_radius_exact": "sqrt(-5 + 3*sqrt(17)/2 + sqrt(409)/2)*sqrt(-3*sqrt(17)/2 + 5 + sqrt(409)/2)*sqrt(-sqrt(409)/2 + 5 + 3*sqrt(17)/2)/sqrt(5 + 3*sqrt(17)/2 + sqrt(409)/2)", "incircle_radius_approx": 2.2538880294148593, "source_dataset": "advanced_geometry", "source_index": 405, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-3, -7), B=(-1, -5), and C=(-10, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.250", "metadata": {"A": ["-3", "-7"], "B": ["-1", "-5"], "C": ["-10", "8"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(2) + sqrt(274)/2)*sqrt(-sqrt(2) + 5*sqrt(10)/2 + sqrt(274)/2)*sqrt(-sqrt(274)/2 + sqrt(2) + 5*sqrt(10)/2)/sqrt(sqrt(2) + 5*sqrt(10)/2 + sqrt(274)/2)", "incircle_radius_approx": 1.2502571273538945, "source_dataset": "advanced_geometry", "source_index": 406, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, -4), B=(2, -7), and C=(9, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.676, -6.649)", "metadata": {"A": ["-4", "-4"], "B": ["2", "-7"], "C": ["9", "8"], "ortho": ["62/37", "-246/37"], "orthocenter_approx": [1.6756756756756757, -6.648648648648648], "source_dataset": "advanced_geometry", "source_index": 407, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 10), B=(-6, 2), and C=(-9, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-29.800, 21.600)", "metadata": {"A": ["5", "10"], "B": ["-6", "2"], "C": ["-9", "-7"], "ortho": ["-149/5", "108/5"], "orthocenter_approx": [-29.8, 21.6], "source_dataset": "advanced_geometry", "source_index": 408, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -6), B=(5, 1), and C=(-9, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.115, -0.951)", "metadata": {"A": ["8", "-6"], "B": ["5", "1"], "C": ["-9", "-7"], "ortho": ["312/61", "-58/61"], "orthocenter_approx": [5.114754098360656, -0.9508196721311475], "source_dataset": "advanced_geometry", "source_index": 409, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, -1), B=(0, -8), and C=(-1, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.456", "metadata": {"A": ["1", "-1"], "B": ["0", "-8"], "C": ["-1", "-8"], "incircle_radius_exact": "sqrt(-1/2 + 5*sqrt(2)/2 + sqrt(53)/2)*sqrt(-5*sqrt(2)/2 + 1/2 + sqrt(53)/2)*sqrt(-sqrt(53)/2 + 1/2 + 5*sqrt(2)/2)/sqrt(1/2 + 5*sqrt(2)/2 + sqrt(53)/2)", "incircle_radius_approx": 0.45599107353681645, "source_dataset": "advanced_geometry", "source_index": 410, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, 7), B=(5, 2), and C=(6, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.000, -0.400)", "metadata": {"A": ["2", "7"], "B": ["5", "2"], "C": ["6", "2"], "ortho": ["2", "-2/5"], "orthocenter_approx": [2.0, -0.4], "source_dataset": "advanced_geometry", "source_index": 411, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 7), B=(10, 8), and C=(0, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.352", "metadata": {"A": ["8", "7"], "B": ["10", "8"], "C": ["0", "-2"], "incircle_radius_exact": "sqrt(-sqrt(145)/2 + sqrt(5)/2 + 5*sqrt(2))*sqrt(-5*sqrt(2) + sqrt(5)/2 + sqrt(145)/2)*sqrt(-sqrt(5)/2 + sqrt(145)/2 + 5*sqrt(2))/sqrt(sqrt(5)/2 + sqrt(145)/2 + 5*sqrt(2))", "incircle_radius_approx": 0.35186738261319683, "source_dataset": "advanced_geometry", "source_index": 412, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, 2), B=(8, 10), and C=(10, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "74.78°", "metadata": {"A": ["-10", "2"], "B": ["8", "10"], "C": ["10", "-3"], "angle_ABC_degrees": 74.78367328797702, "source_dataset": "advanced_geometry", "source_index": 413, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, 4), B=(3, 2), and C=(10, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "120.96°", "metadata": {"A": ["4", "4"], "B": ["3", "2"], "C": ["10", "-9"], "angle_ABC_degrees": 120.96375653207352, "source_dataset": "advanced_geometry", "source_index": 414, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 7), B=(-2, 4), and C=(10, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.091, 6.727)", "metadata": {"A": ["0", "7"], "B": ["-2", "4"], "C": ["10", "0"], "ortho": ["-1/11", "74/11"], "orthocenter_approx": [-0.09090909090909091, 6.7272727272727275], "source_dataset": "advanced_geometry", "source_index": 415, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 6), B=(4, -8), and C=(-8, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.690, -4.818)", "metadata": {"A": ["9", "6"], "B": ["4", "-8"], "C": ["-8", "-1"], "ortho": ["78/29", "-978/203"], "orthocenter_approx": [2.689655172413793, -4.817733990147783], "source_dataset": "advanced_geometry", "source_index": 416, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, 2), B=(-10, -5), and C=(4, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "33.69°", "metadata": {"A": ["-6", "2"], "B": ["-10", "-5"], "C": ["4", "2"], "angle_ABC_degrees": 33.690067525979785, "source_dataset": "advanced_geometry", "source_index": 417, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, -10), B=(2, 9), and C=(0, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.192", "metadata": {"A": ["5", "-10"], "B": ["2", "9"], "C": ["0", "-10"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(365)/2 + sqrt(370)/2)*sqrt(-sqrt(365)/2 + 5/2 + sqrt(370)/2)*sqrt(-sqrt(370)/2 + 5/2 + sqrt(365)/2)/sqrt(5/2 + sqrt(365)/2 + sqrt(370)/2)", "incircle_radius_approx": 2.191952398597682, "source_dataset": "advanced_geometry", "source_index": 418, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(0, -8), B=(8, 9), and C=(7, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.80°", "metadata": {"A": ["0", "-8"], "B": ["8", "9"], "C": ["7", "8"], "angle_ABC_degrees": 19.798876354524932, "source_dataset": "advanced_geometry", "source_index": 419, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-8, 2), B=(-5, -10), and C=(-7, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "75.96°", "metadata": {"A": ["-8", "2"], "B": ["-5", "-10"], "C": ["-7", "-10"], "angle_ABC_degrees": 75.96375653207353, "source_dataset": "advanced_geometry", "source_index": 420, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, 7), B=(-8, -4), and C=(6, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.049, -3.330)", "metadata": {"A": ["5", "7"], "B": ["-8", "-4"], "C": ["6", "-8"], "ortho": ["211/103", "-343/103"], "orthocenter_approx": [2.0485436893203883, -3.3300970873786406], "source_dataset": "advanced_geometry", "source_index": 421, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 7), B=(9, -3), and C=(-3, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.756", "metadata": {"A": ["-5", "7"], "B": ["9", "-3"], "C": ["-3", "-2"], "incircle_radius_exact": "sqrt(-sqrt(145)/2 + sqrt(85)/2 + sqrt(74))*sqrt(-sqrt(74) + sqrt(85)/2 + sqrt(145)/2)*sqrt(-sqrt(85)/2 + sqrt(145)/2 + sqrt(74))/sqrt(sqrt(85)/2 + sqrt(145)/2 + sqrt(74))", "incircle_radius_approx": 2.7556954162251537, "source_dataset": "advanced_geometry", "source_index": 422, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, -9), B=(10, -10), and C=(-5, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.947, -15.316)", "metadata": {"A": ["-3", "-9"], "B": ["10", "-10"], "C": ["-5", "-3"], "ortho": ["-113/19", "-291/19"], "orthocenter_approx": [-5.947368421052632, -15.31578947368421], "source_dataset": "advanced_geometry", "source_index": 423, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, -1), B=(-4, -6), and C=(-6, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-46.000, -27.000)", "metadata": {"A": ["-7", "-1"], "B": ["-4", "-6"], "C": ["-6", "-3"], "ortho": ["-46", "-27"], "orthocenter_approx": [-46.0, -27.0], "source_dataset": "advanced_geometry", "source_index": 424, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, 2), B=(-8, -10), and C=(3, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.427", "metadata": {"A": ["5", "2"], "B": ["-8", "-10"], "C": ["3", "-7"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + sqrt(85)/2 + sqrt(313)/2)*sqrt(-sqrt(313)/2 + sqrt(85)/2 + sqrt(130)/2)*sqrt(-sqrt(85)/2 + sqrt(130)/2 + sqrt(313)/2)/sqrt(sqrt(85)/2 + sqrt(130)/2 + sqrt(313)/2)", "incircle_radius_approx": 2.4273678856531453, "source_dataset": "advanced_geometry", "source_index": 425, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, -9), B=(8, -8), and C=(-5, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.421, -11.471)", "metadata": {"A": ["2", "-9"], "B": ["8", "-8"], "C": ["-5", "10"], "ortho": ["-172/121", "-1388/121"], "orthocenter_approx": [-1.421487603305785, -11.47107438016529], "source_dataset": "advanced_geometry", "source_index": 426, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, -2), B=(3, 10), and C=(2, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.461", "metadata": {"A": ["-1", "-2"], "B": ["3", "10"], "C": ["2", "10"], "incircle_radius_exact": "sqrt(-1/2 + 3*sqrt(17)/2 + 2*sqrt(10))*sqrt(-2*sqrt(10) + 1/2 + 3*sqrt(17)/2)*sqrt(-3*sqrt(17)/2 + 1/2 + 2*sqrt(10))/sqrt(1/2 + 3*sqrt(17)/2 + 2*sqrt(10))", "incircle_radius_approx": 0.4612115775219919, "source_dataset": "advanced_geometry", "source_index": 427, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, -5), B=(-6, 6), and C=(10, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.278, -0.550)", "metadata": {"A": ["1", "-5"], "B": ["-6", "6"], "C": ["10", "5"], "ortho": ["216/169", "-93/169"], "orthocenter_approx": [1.2781065088757397, -0.5502958579881657], "source_dataset": "advanced_geometry", "source_index": 428, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, 10), B=(-1, 4), and C=(0, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "149.04°", "metadata": {"A": ["-8", "10"], "B": ["-1", "4"], "C": ["0", "1"], "angle_ABC_degrees": 149.03624346792648, "source_dataset": "advanced_geometry", "source_index": 429, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 10), B=(4, -9), and C=(-5, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.065, -8.290)", "metadata": {"A": ["-4", "10"], "B": ["4", "-9"], "C": ["-5", "-7"], "ortho": ["-250/31", "-257/31"], "orthocenter_approx": [-8.064516129032258, -8.290322580645162], "source_dataset": "advanced_geometry", "source_index": 430, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -9), B=(-10, 1), and C=(-7, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.352", "metadata": {"A": ["0", "-9"], "B": ["-10", "1"], "C": ["-7", "-1"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + sqrt(13)/2 + 5*sqrt(2))*sqrt(-sqrt(13)/2 + sqrt(113)/2 + 5*sqrt(2))*sqrt(-5*sqrt(2) + sqrt(13)/2 + sqrt(113)/2)/sqrt(sqrt(13)/2 + sqrt(113)/2 + 5*sqrt(2))", "incircle_radius_approx": 0.3523877281789796, "source_dataset": "advanced_geometry", "source_index": 431, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 3), B=(-6, -1), and C=(1, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.765, -4.824)", "metadata": {"A": ["-9", "3"], "B": ["-6", "-1"], "C": ["1", "1"], "ortho": ["-115/17", "-82/17"], "orthocenter_approx": [-6.764705882352941, -4.823529411764706], "source_dataset": "advanced_geometry", "source_index": 432, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 10), B=(7, 4), and C=(0, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.066, 1.686)", "metadata": {"A": ["-10", "10"], "B": ["7", "4"], "C": ["0", "-7"], "ortho": ["702/229", "386/229"], "orthocenter_approx": [3.0655021834061134, 1.685589519650655], "source_dataset": "advanced_geometry", "source_index": 433, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, -2), B=(1, 9), and C=(1, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.284", "metadata": {"A": ["8", "-2"], "B": ["1", "9"], "C": ["1", "4"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(85)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + 5/2 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + 5/2 + sqrt(170)/2)/sqrt(5/2 + sqrt(85)/2 + sqrt(170)/2)", "incircle_radius_approx": 1.284029097576921, "source_dataset": "advanced_geometry", "source_index": 434, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, -9), B=(-1, -5), and C=(7, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "103.39°", "metadata": {"A": ["3", "-9"], "B": ["-1", "-5"], "C": ["7", "8"], "angle_ABC_degrees": 103.3924977537511, "source_dataset": "advanced_geometry", "source_index": 435, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -7), B=(6, -10), and C=(-1, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.365", "metadata": {"A": ["-6", "-7"], "B": ["6", "-10"], "C": ["-1", "2"], "incircle_radius_exact": "sqrt(-sqrt(106)/2 + 3*sqrt(17)/2 + sqrt(193)/2)*sqrt(-3*sqrt(17)/2 + sqrt(106)/2 + sqrt(193)/2)*sqrt(-sqrt(193)/2 + sqrt(106)/2 + 3*sqrt(17)/2)/sqrt(sqrt(106)/2 + 3*sqrt(17)/2 + sqrt(193)/2)", "incircle_radius_approx": 3.3645727063912463, "source_dataset": "advanced_geometry", "source_index": 436, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, 6), B=(6, 6), and C=(10, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.121", "metadata": {"A": ["7", "6"], "B": ["6", "6"], "C": ["10", "7"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(10)/2 + sqrt(17)/2)*sqrt(-sqrt(10)/2 + 1/2 + sqrt(17)/2)*sqrt(-sqrt(17)/2 + 1/2 + sqrt(10)/2)/sqrt(1/2 + sqrt(10)/2 + sqrt(17)/2)", "incircle_radius_approx": 0.12069447670761912, "source_dataset": "advanced_geometry", "source_index": 437, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(8, -1), B=(6, 4), and C=(-4, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.100", "metadata": {"A": ["8", "-1"], "B": ["6", "4"], "C": ["-4", "-1"], "incircle_radius_exact": "sqrt(-6 + sqrt(29)/2 + 5*sqrt(5)/2)*sqrt(-sqrt(29)/2 + 5*sqrt(5)/2 + 6)*sqrt(-5*sqrt(5)/2 + sqrt(29)/2 + 6)/sqrt(sqrt(29)/2 + 5*sqrt(5)/2 + 6)", "incircle_radius_approx": 2.100435495238146, "source_dataset": "advanced_geometry", "source_index": 438, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 3), B=(-6, 4), and C=(10, -1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "12.16°", "metadata": {"A": ["5", "3"], "B": ["-6", "4"], "C": ["10", "-1"], "angle_ABC_degrees": 12.159595728526517, "source_dataset": "advanced_geometry", "source_index": 439, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, -1), B=(3, 10), and C=(8, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.43°", "metadata": {"A": ["9", "-1"], "B": ["3", "10"], "C": ["8", "7"], "angle_ABC_degrees": 30.42578380196126, "source_dataset": "advanced_geometry", "source_index": 440, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, 0), B=(-2, -8), and C=(-10, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "148.47°", "metadata": {"A": ["8", "0"], "B": ["-2", "-8"], "C": ["-10", "-9"], "angle_ABC_degrees": 148.46520809481171, "source_dataset": "advanced_geometry", "source_index": 441, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, 4), B=(0, -9), and C=(-7, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "7.26°", "metadata": {"A": ["-5", "4"], "B": ["0", "-9"], "C": ["-7", "4"], "angle_ABC_degrees": 7.263244740584561, "source_dataset": "advanced_geometry", "source_index": 442, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -5), B=(5, 0), and C=(1, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.667, 5.667)", "metadata": {"A": ["-6", "-5"], "B": ["5", "0"], "C": ["1", "2"], "ortho": ["-2/3", "17/3"], "orthocenter_approx": [-0.6666666666666666, 5.666666666666667], "source_dataset": "advanced_geometry", "source_index": 443, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-5, 5), B=(-2, -5), and C=(-3, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.000, 5.300)", "metadata": {"A": ["-5", "5"], "B": ["-2", "-5"], "C": ["-3", "5"], "ortho": ["-2", "53/10"], "orthocenter_approx": [-2.0, 5.3], "source_dataset": "advanced_geometry", "source_index": 444, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, 0), B=(-9, -5), and C=(-2, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.97°", "metadata": {"A": ["1", "0"], "B": ["-9", "-5"], "C": ["-2", "0"], "angle_ABC_degrees": 8.972626614896393, "source_dataset": "advanced_geometry", "source_index": 445, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 0), B=(6, 10), and C=(7, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(22.000, 2.000)", "metadata": {"A": ["6", "0"], "B": ["6", "10"], "C": ["7", "2"], "ortho": ["22", "2"], "orthocenter_approx": [22.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 446, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, -9), B=(7, 1), and C=(-10, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.059, 1.000)", "metadata": {"A": ["-10", "-9"], "B": ["7", "1"], "C": ["-10", "6"], "ortho": ["-120/17", "1"], "orthocenter_approx": [-7.0588235294117645, 1.0], "source_dataset": "advanced_geometry", "source_index": 447, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -1), B=(5, -3), and C=(6, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "58.74°", "metadata": {"A": ["10", "-1"], "B": ["5", "-3"], "C": ["6", "3"], "angle_ABC_degrees": 58.73626830562257, "source_dataset": "advanced_geometry", "source_index": 448, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 5), B=(9, 10), and C=(8, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.885", "metadata": {"A": ["2", "5"], "B": ["9", "10"], "C": ["8", "7"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(74)/2)*sqrt(sqrt(10)/2 + sqrt(74)/2)*sqrt(-sqrt(74)/2 + 3*sqrt(10)/2)/sqrt(sqrt(74)/2 + 3*sqrt(10)/2)", "incircle_radius_approx": 0.8845077134625112, "source_dataset": "advanced_geometry", "source_index": 449, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, 1), B=(-1, -10), and C=(-9, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "6.81°", "metadata": {"A": ["-7", "1"], "B": ["-1", "-10"], "C": ["-9", "10"], "angle_ABC_degrees": 6.809050179613405, "source_dataset": "advanced_geometry", "source_index": 450, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, 8), B=(-2, 1), and C=(6, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.886", "metadata": {"A": ["-1", "8"], "B": ["-2", "1"], "C": ["6", "9"], "incircle_radius_exact": "4*sqrt(2)/3", "incircle_radius_approx": 1.8856180831641267, "source_dataset": "advanced_geometry", "source_index": 451, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 10), B=(-6, -10), and C=(-10, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.18°", "metadata": {"A": ["9", "10"], "B": ["-6", "-10"], "C": ["-10", "10"], "angle_ABC_degrees": 48.179830119864235, "source_dataset": "advanced_geometry", "source_index": 452, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, -4), B=(7, -7), and C=(8, -4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "94.40°", "metadata": {"A": ["-5", "-4"], "B": ["7", "-7"], "C": ["8", "-4"], "angle_ABC_degrees": 94.39870535499553, "source_dataset": "advanced_geometry", "source_index": 453, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, 2), B=(-7, 0), and C=(10, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "20.31°", "metadata": {"A": ["4", "2"], "B": ["-7", "0"], "C": ["10", "-3"], "angle_ABC_degrees": 20.312826270207367, "source_dataset": "advanced_geometry", "source_index": 454, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, -5), B=(0, -4), and C=(-1, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.387", "metadata": {"A": ["9", "-5"], "B": ["0", "-4"], "C": ["-1", "-3"], "incircle_radius_exact": "sqrt(-sqrt(2)/2 + sqrt(82)/2 + sqrt(26))*sqrt(-sqrt(26) + sqrt(2)/2 + sqrt(82)/2)*sqrt(-sqrt(82)/2 + sqrt(2)/2 + sqrt(26))/sqrt(sqrt(2)/2 + sqrt(82)/2 + sqrt(26))", "incircle_radius_approx": 0.3870785866001241, "source_dataset": "advanced_geometry", "source_index": 455, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, 9), B=(-3, -3), and C=(1, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.664", "metadata": {"A": ["3", "9"], "B": ["-3", "-3"], "C": ["1", "2"], "incircle_radius_exact": "sqrt(-sqrt(41)/2 + sqrt(53)/2 + 3*sqrt(5))*sqrt(-3*sqrt(5) + sqrt(41)/2 + sqrt(53)/2)*sqrt(-sqrt(53)/2 + sqrt(41)/2 + 3*sqrt(5))/sqrt(sqrt(41)/2 + sqrt(53)/2 + 3*sqrt(5))", "incircle_radius_approx": 0.664215416775799, "source_dataset": "advanced_geometry", "source_index": 456, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 5), B=(-1, 8), and C=(5, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.724, 6.966)", "metadata": {"A": ["-5", "5"], "B": ["-1", "8"], "C": ["5", "-2"], "ortho": ["-50/29", "202/29"], "orthocenter_approx": [-1.7241379310344827, 6.9655172413793105], "source_dataset": "advanced_geometry", "source_index": 457, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, 8), B=(4, -2), and C=(10, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.112", "metadata": {"A": ["6", "8"], "B": ["4", "-2"], "C": ["10", "2"], "incircle_radius_exact": "sqrt(-sqrt(26) + 2*sqrt(13))*sqrt(26*sqrt(26) + 52*sqrt(13))/(sqrt(26) + 2*sqrt(13))", "incircle_radius_approx": 2.1120830373351938, "source_dataset": "advanced_geometry", "source_index": 458, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 10), B=(1, -3), and C=(10, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "80.31°", "metadata": {"A": ["-1", "10"], "B": ["1", "-3"], "C": ["10", "0"], "angle_ABC_degrees": 80.3112134396332, "source_dataset": "advanced_geometry", "source_index": 459, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, -5), B=(10, 1), and C=(-5, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "41.28°", "metadata": {"A": ["1", "-5"], "B": ["10", "1"], "C": ["-5", "3"], "angle_ABC_degrees": 41.28471089457123, "source_dataset": "advanced_geometry", "source_index": 460, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-6, 1), B=(-4, 2), and C=(-1, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.737, 1.474)", "metadata": {"A": ["-6", "1"], "B": ["-4", "2"], "C": ["-1", "-6"], "ortho": ["-90/19", "28/19"], "orthocenter_approx": [-4.7368421052631575, 1.4736842105263157], "source_dataset": "advanced_geometry", "source_index": 461, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(8, 1), B=(-10, -4), and C=(-9, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.043", "metadata": {"A": ["8", "1"], "B": ["-10", "-4"], "C": ["-9", "-6"], "incircle_radius_exact": "sqrt(-13*sqrt(2)/2 + sqrt(5)/2 + sqrt(349)/2)*sqrt(-sqrt(349)/2 + sqrt(5)/2 + 13*sqrt(2)/2)*sqrt(-sqrt(5)/2 + 13*sqrt(2)/2 + sqrt(349)/2)/sqrt(sqrt(5)/2 + 13*sqrt(2)/2 + sqrt(349)/2)", "incircle_radius_approx": 1.0431936631078247, "source_dataset": "advanced_geometry", "source_index": 462, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -1), B=(-2, 0), and C=(5, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.000, -1.000)", "metadata": {"A": ["-2", "-1"], "B": ["-2", "0"], "C": ["5", "-1"], "ortho": ["-2", "-1"], "orthocenter_approx": [-2.0, -1.0], "source_dataset": "advanced_geometry", "source_index": 463, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, 2), B=(8, 3), and C=(-2, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.872", "metadata": {"A": ["-2", "2"], "B": ["8", "3"], "C": ["-2", "-3"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(101)/2 + sqrt(34))*sqrt(-sqrt(101)/2 + 5/2 + sqrt(34))*sqrt(-sqrt(34) + 5/2 + sqrt(101)/2)/sqrt(5/2 + sqrt(101)/2 + sqrt(34))", "incircle_radius_approx": 1.8718333672582925, "source_dataset": "advanced_geometry", "source_index": 464, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 8), B=(-6, 5), and C=(10, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.798", "metadata": {"A": ["9", "8"], "B": ["-6", "5"], "C": ["10", "-3"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + 3*sqrt(26)/2 + 4*sqrt(5))*sqrt(-3*sqrt(26)/2 + sqrt(122)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(122)/2 + 3*sqrt(26)/2)/sqrt(sqrt(122)/2 + 3*sqrt(26)/2 + 4*sqrt(5))", "incircle_radius_approx": 3.7982441974957832, "source_dataset": "advanced_geometry", "source_index": 465, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(3, 5), B=(-8, -7), and C=(-2, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "5.64°", "metadata": {"A": ["3", "5"], "B": ["-8", "-7"], "C": ["-2", "1"], "angle_ABC_degrees": 5.640549432156822, "source_dataset": "advanced_geometry", "source_index": 466, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, -4), B=(-5, -6), and C=(-7, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "84.09°", "metadata": {"A": ["3", "-4"], "B": ["-5", "-6"], "C": ["-7", "8"], "angle_ABC_degrees": 84.0938588862295, "source_dataset": "advanced_geometry", "source_index": 467, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(7, -1), B=(3, 3), and C=(-7, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "87.27°", "metadata": {"A": ["7", "-1"], "B": ["3", "3"], "C": ["-7", "-8"], "angle_ABC_degrees": 87.27368900609375, "source_dataset": "advanced_geometry", "source_index": 468, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, -7), B=(-6, -3), and C=(5, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.059, -6.353)", "metadata": {"A": ["4", "-7"], "B": ["-6", "-3"], "C": ["5", "-4"], "ortho": ["69/17", "-108/17"], "orthocenter_approx": [4.0588235294117645, -6.352941176470588], "source_dataset": "advanced_geometry", "source_index": 469, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, -6), B=(0, 3), and C=(1, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-22.727, -1.545)", "metadata": {"A": ["4", "-6"], "B": ["0", "3"], "C": ["1", "9"], "ortho": ["-250/11", "-17/11"], "orthocenter_approx": [-22.727272727272727, -1.5454545454545454], "source_dataset": "advanced_geometry", "source_index": 470, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -6), B=(9, -5), and C=(3, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.800, -2.400)", "metadata": {"A": ["-4", "-6"], "B": ["9", "-5"], "C": ["3", "8"], "ortho": ["19/5", "-12/5"], "orthocenter_approx": [3.8, -2.4], "source_dataset": "advanced_geometry", "source_index": 471, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 10), B=(10, -4), and C=(-6, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.255", "metadata": {"A": ["-7", "10"], "B": ["10", "-4"], "C": ["-6", "3"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(305)/2 + sqrt(485)/2)*sqrt(-sqrt(305)/2 + 5*sqrt(2)/2 + sqrt(485)/2)*sqrt(-sqrt(485)/2 + 5*sqrt(2)/2 + sqrt(305)/2)/sqrt(5*sqrt(2)/2 + sqrt(305)/2 + sqrt(485)/2)", "incircle_radius_approx": 2.2552499373390247, "source_dataset": "advanced_geometry", "source_index": 472, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -1), B=(-6, -7), and C=(-10, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.324, -1.118)", "metadata": {"A": ["-2", "-1"], "B": ["-6", "-7"], "C": ["-10", "4"], "ortho": ["-79/34", "-19/17"], "orthocenter_approx": [-2.323529411764706, -1.1176470588235294], "source_dataset": "advanced_geometry", "source_index": 473, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-3, 7), B=(8, 10), and C=(6, -10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "69.03°", "metadata": {"A": ["-3", "7"], "B": ["8", "10"], "C": ["6", "-10"], "angle_ABC_degrees": 69.03428815944258, "source_dataset": "advanced_geometry", "source_index": 474, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -4), B=(3, -5), and C=(5, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.592", "metadata": {"A": ["2", "-4"], "B": ["3", "-5"], "C": ["5", "-1"], "incircle_radius_exact": "sqrt(-sqrt(2) + sqrt(5))*sqrt(sqrt(2) + sqrt(5))*sqrt(-sqrt(5) + 2*sqrt(2))/sqrt(sqrt(5) + 2*sqrt(2))", "incircle_radius_approx": 0.5923591472464004, "source_dataset": "advanced_geometry", "source_index": 475, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 1), B=(-2, 8), and C=(-2, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.725", "metadata": {"A": ["6", "1"], "B": ["-2", "8"], "C": ["-2", "6"], "incircle_radius_exact": "sqrt(-1 + sqrt(89)/2 + sqrt(113)/2)*sqrt(-sqrt(113)/2 + 1 + sqrt(89)/2)*sqrt(-sqrt(89)/2 + 1 + sqrt(113)/2)/sqrt(1 + sqrt(89)/2 + sqrt(113)/2)", "incircle_radius_approx": 0.7251589895233616, "source_dataset": "advanced_geometry", "source_index": 476, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, -10), B=(0, -3), and C=(-6, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "138.18°", "metadata": {"A": ["4", "-10"], "B": ["0", "-3"], "C": ["-6", "-1"], "angle_ABC_degrees": 138.17983011986422, "source_dataset": "advanced_geometry", "source_index": 477, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, 2), B=(2, -6), and C=(4, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.565, -0.348)", "metadata": {"A": ["-9", "2"], "B": ["2", "-6"], "C": ["4", "3"], "ortho": ["36/23", "-8/23"], "orthocenter_approx": [1.565217391304348, -0.34782608695652173], "source_dataset": "advanced_geometry", "source_index": 478, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, -1), B=(-2, 3), and C=(-5, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(14.000, -37.000)", "metadata": {"A": ["-10", "-1"], "B": ["-2", "3"], "C": ["-5", "1"], "ortho": ["14", "-37"], "orthocenter_approx": [14.0, -37.0], "source_dataset": "advanced_geometry", "source_index": 479, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, 2), B=(9, -10), and C=(-6, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.077, -6.077)", "metadata": {"A": ["-3", "2"], "B": ["9", "-10"], "C": ["-6", "-8"], "ortho": ["-53/13", "-79/13"], "orthocenter_approx": [-4.076923076923077, -6.076923076923077], "source_dataset": "advanced_geometry", "source_index": 480, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-1, 6), B=(2, -5), and C=(5, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "105.26°", "metadata": {"A": ["-1", "6"], "B": ["2", "-5"], "C": ["5", "-5"], "angle_ABC_degrees": 105.25511870305778, "source_dataset": "advanced_geometry", "source_index": 481, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, 8), B=(-2, 5), and C=(0, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.980, 10.612)", "metadata": {"A": ["-10", "8"], "B": ["-2", "5"], "C": ["0", "-8"], "ortho": ["342/49", "520/49"], "orthocenter_approx": [6.979591836734694, 10.612244897959183], "source_dataset": "advanced_geometry", "source_index": 482, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, 5), B=(-5, -5), and C=(9, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "109.44°", "metadata": {"A": ["-7", "5"], "B": ["-5", "-5"], "C": ["9", "-7"], "angle_ABC_degrees": 109.44003482817618, "source_dataset": "advanced_geometry", "source_index": 483, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -5), B=(-9, -1), and C=(2, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.898, -5.373)", "metadata": {"A": ["-4", "-5"], "B": ["-9", "-1"], "C": ["2", "2"], "ortho": ["-230/59", "-317/59"], "orthocenter_approx": [-3.8983050847457625, -5.372881355932203], "source_dataset": "advanced_geometry", "source_index": 484, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, -5), B=(0, -3), and C=(-1, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.083", "metadata": {"A": ["0", "-5"], "B": ["0", "-3"], "C": ["-1", "7"], "incircle_radius_exact": "sqrt(-1 + sqrt(101)/2 + sqrt(145)/2)*sqrt(-sqrt(101)/2 + 1 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + 1 + sqrt(101)/2)/sqrt(1 + sqrt(101)/2 + sqrt(145)/2)", "incircle_radius_approx": 0.08301693435078142, "source_dataset": "advanced_geometry", "source_index": 485, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -1), B=(-7, 5), and C=(5, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.462, 9.154)", "metadata": {"A": ["8", "-1"], "B": ["-7", "5"], "C": ["5", "8"], "ortho": ["71/13", "119/13"], "orthocenter_approx": [5.461538461538462, 9.153846153846153], "source_dataset": "advanced_geometry", "source_index": 486, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, -3), B=(4, -4), and C=(3, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "39.29°", "metadata": {"A": ["-6", "-3"], "B": ["4", "-4"], "C": ["3", "-3"], "angle_ABC_degrees": 39.28940686250036, "source_dataset": "advanced_geometry", "source_index": 487, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 7), B=(-3, 9), and C=(5, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.978", "metadata": {"A": ["-7", "7"], "B": ["-3", "9"], "C": ["5", "-8"], "incircle_radius_exact": "sqrt(-sqrt(353)/2 + sqrt(5) + 3*sqrt(41)/2)*sqrt(-3*sqrt(41)/2 + sqrt(5) + sqrt(353)/2)*sqrt(-sqrt(5) + sqrt(353)/2 + 3*sqrt(41)/2)/sqrt(sqrt(5) + sqrt(353)/2 + 3*sqrt(41)/2)", "incircle_radius_approx": 1.9778759088422586, "source_dataset": "advanced_geometry", "source_index": 488, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 4), B=(8, 1), and C=(3, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "38.10°", "metadata": {"A": ["-6", "4"], "B": ["8", "1"], "C": ["3", "7"], "angle_ABC_degrees": 38.099671830722706, "source_dataset": "advanced_geometry", "source_index": 489, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, 9), B=(-6, -8), and C=(5, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(41.593, -34.441)", "metadata": {"A": ["10", "9"], "B": ["-6", "-8"], "C": ["5", "0"], "ortho": ["2454/59", "-2032/59"], "orthocenter_approx": [41.59322033898305, -34.440677966101696], "source_dataset": "advanced_geometry", "source_index": 490, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 1), B=(-2, 3), and C=(-9, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.209", "metadata": {"A": ["-5", "1"], "B": ["-2", "3"], "C": ["-9", "5"], "incircle_radius_exact": "sqrt(-sqrt(13)/2 + 2*sqrt(2) + sqrt(53)/2)*sqrt(-2*sqrt(2) + sqrt(13)/2 + sqrt(53)/2)*sqrt(-sqrt(53)/2 + sqrt(13)/2 + 2*sqrt(2))/sqrt(sqrt(13)/2 + 2*sqrt(2) + sqrt(53)/2)", "incircle_radius_approx": 1.209005976369683, "source_dataset": "advanced_geometry", "source_index": 491, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -4), B=(4, 6), and C=(-6, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.407", "metadata": {"A": ["7", "-4"], "B": ["4", "6"], "C": ["-6", "-2"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + sqrt(41) + sqrt(173)/2)*sqrt(-sqrt(173)/2 + sqrt(109)/2 + sqrt(41))*sqrt(-sqrt(41) + sqrt(109)/2 + sqrt(173)/2)/sqrt(sqrt(109)/2 + sqrt(41) + sqrt(173)/2)", "incircle_radius_approx": 3.406640068040391, "source_dataset": "advanced_geometry", "source_index": 492, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, 2), B=(8, 2), and C=(-6, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.000, 16.000)", "metadata": {"A": ["3", "2"], "B": ["8", "2"], "C": ["-6", "-7"], "ortho": ["-6", "16"], "orthocenter_approx": [-6.0, 16.0], "source_dataset": "advanced_geometry", "source_index": 493, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 10), B=(7, 7), and C=(-10, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "15.26°", "metadata": {"A": ["-4", "10"], "B": ["7", "7"], "C": ["-10", "7"], "angle_ABC_degrees": 15.255118703057777, "source_dataset": "advanced_geometry", "source_index": 494, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, -3), B=(-3, -9), and C=(4, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "24.38°", "metadata": {"A": ["4", "-3"], "B": ["-3", "-9"], "C": ["4", "6"], "angle_ABC_degrees": 24.38181187689551, "source_dataset": "advanced_geometry", "source_index": 495, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -9), B=(2, -9), and C=(-5, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.002", "metadata": {"A": ["-1", "-9"], "B": ["2", "-9"], "C": ["-5", "-2"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(65)/2 + 7*sqrt(2)/2)*sqrt(-7*sqrt(2)/2 + 3/2 + sqrt(65)/2)*sqrt(-sqrt(65)/2 + 3/2 + 7*sqrt(2)/2)/sqrt(3/2 + sqrt(65)/2 + 7*sqrt(2)/2)", "incircle_radius_approx": 1.001824623907395, "source_dataset": "advanced_geometry", "source_index": 496, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, 7), B=(-9, -6), and C=(1, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.235", "metadata": {"A": ["-4", "7"], "B": ["-9", "-6"], "C": ["1", "-3"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + 5*sqrt(5)/2 + sqrt(194)/2)*sqrt(-sqrt(194)/2 + sqrt(109)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + sqrt(109)/2 + sqrt(194)/2)/sqrt(sqrt(109)/2 + 5*sqrt(5)/2 + sqrt(194)/2)", "incircle_radius_approx": 3.2349682925546164, "source_dataset": "advanced_geometry", "source_index": 497, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 1), B=(-10, 4), and C=(0, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "35.54°", "metadata": {"A": ["9", "1"], "B": ["-10", "4"], "C": ["0", "9"], "angle_ABC_degrees": 35.53767779197438, "source_dataset": "advanced_geometry", "source_index": 498, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 4), B=(2, -6), and C=(-5, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.375", "metadata": {"A": ["-10", "4"], "B": ["2", "-6"], "C": ["-5", "7"], "incircle_radius_exact": "sqrt(-sqrt(218)/2 + sqrt(34)/2 + sqrt(61))*sqrt(-sqrt(34)/2 + sqrt(218)/2 + sqrt(61))*sqrt(-sqrt(61) + sqrt(34)/2 + sqrt(218)/2)/sqrt(sqrt(34)/2 + sqrt(218)/2 + sqrt(61))", "incircle_radius_approx": 2.3746230567851114, "source_dataset": "advanced_geometry", "source_index": 499, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -5), B=(3, -4), and C=(-5, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.877", "metadata": {"A": ["-7", "-5"], "B": ["3", "-4"], "C": ["-5", "4"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(85)/2 + 4*sqrt(2))*sqrt(-4*sqrt(2) + sqrt(85)/2 + sqrt(101)/2)*sqrt(-sqrt(85)/2 + sqrt(101)/2 + 4*sqrt(2))/sqrt(sqrt(85)/2 + sqrt(101)/2 + 4*sqrt(2))", "incircle_radius_approx": 2.8774034604502017, "source_dataset": "advanced_geometry", "source_index": 500, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -8), B=(4, 9), and C=(-5, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.82°", "metadata": {"A": ["-6", "-8"], "B": ["4", "9"], "C": ["-5", "-2"], "angle_ABC_degrees": 8.823861943040479, "source_dataset": "advanced_geometry", "source_index": 501, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, -7), B=(-1, 9), and C=(7, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "39.60°", "metadata": {"A": ["-2", "-7"], "B": ["-1", "9"], "C": ["7", "-2"], "angle_ABC_degrees": 39.60370776010096, "source_dataset": "advanced_geometry", "source_index": 502, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, 10), B=(-9, -3), and C=(5, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.061, -6.628)", "metadata": {"A": ["10", "10"], "B": ["-9", "-3"], "C": ["5", "-8"], "ortho": ["1125/277", "-1836/277"], "orthocenter_approx": [4.061371841155235, -6.628158844765343], "source_dataset": "advanced_geometry", "source_index": 503, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-2, 2), B=(2, -9), and C=(1, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.000, 2.364)", "metadata": {"A": ["-2", "2"], "B": ["2", "-9"], "C": ["1", "2"], "ortho": ["2", "26/11"], "orthocenter_approx": [2.0, 2.3636363636363638], "source_dataset": "advanced_geometry", "source_index": 504, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, -2), B=(0, 5), and C=(-4, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-19.167, 20.333)", "metadata": {"A": ["-8", "-2"], "B": ["0", "5"], "C": ["-4", "3"], "ortho": ["-115/6", "61/3"], "orthocenter_approx": [-19.166666666666668, 20.333333333333332], "source_dataset": "advanced_geometry", "source_index": 505, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 6), B=(-6, -8), and C=(2, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.000, 4.286)", "metadata": {"A": ["-9", "6"], "B": ["-6", "-8"], "C": ["2", "6"], "ortho": ["-6", "30/7"], "orthocenter_approx": [-6.0, 4.285714285714286], "source_dataset": "advanced_geometry", "source_index": 506, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 5), B=(-5, 7), and C=(5, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "13.24°", "metadata": {"A": ["-4", "5"], "B": ["-5", "7"], "C": ["5", "-5"], "angle_ABC_degrees": 13.240519915187205, "source_dataset": "advanced_geometry", "source_index": 507, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, -4), B=(1, -9), and C=(-3, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.442", "metadata": {"A": ["5", "-4"], "B": ["1", "-9"], "C": ["-3", "-7"], "incircle_radius_exact": "sqrt(-sqrt(41)/2 + sqrt(5) + sqrt(73)/2)*sqrt(-sqrt(5) + sqrt(41)/2 + sqrt(73)/2)*sqrt(-sqrt(73)/2 + sqrt(5) + sqrt(41)/2)/sqrt(sqrt(5) + sqrt(41)/2 + sqrt(73)/2)", "incircle_radius_approx": 1.44186721441947, "source_dataset": "advanced_geometry", "source_index": 508, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -3), B=(-10, -4), and C=(-3, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "32.68°", "metadata": {"A": ["10", "-3"], "B": ["-10", "-4"], "C": ["-3", "1"], "angle_ABC_degrees": 32.675272565862635, "source_dataset": "advanced_geometry", "source_index": 509, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, 6), B=(2, -9), and C=(-3, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(26.571, 21.714)", "metadata": {"A": ["-8", "6"], "B": ["2", "-9"], "C": ["-3", "2"], "ortho": ["186/7", "152/7"], "orthocenter_approx": [26.571428571428573, 21.714285714285715], "source_dataset": "advanced_geometry", "source_index": 510, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 4), B=(-6, 4), and C=(-10, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.438", "metadata": {"A": ["-10", "4"], "B": ["-6", "4"], "C": ["-10", "3"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(17)/2)*sqrt(3/2 + sqrt(17)/2)*sqrt(5/2 - sqrt(17)/2)/sqrt(sqrt(17)/2 + 5/2)", "incircle_radius_approx": 0.4384471871911697, "source_dataset": "advanced_geometry", "source_index": 511, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, -5), B=(1, -4), and C=(2, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "139.09°", "metadata": {"A": ["0", "-5"], "B": ["1", "-4"], "C": ["2", "10"], "angle_ABC_degrees": 139.08561677997488, "source_dataset": "advanced_geometry", "source_index": 512, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -5), B=(-6, 7), and C=(-7, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.625, -4.875)", "metadata": {"A": ["-2", "-5"], "B": ["-6", "7"], "C": ["-7", "-6"], "ortho": ["-29/8", "-39/8"], "orthocenter_approx": [-3.625, -4.875], "source_dataset": "advanced_geometry", "source_index": 513, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, -4), B=(10, 8), and C=(4, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.95°", "metadata": {"A": ["5", "-4"], "B": ["10", "8"], "C": ["4", "6"], "angle_ABC_degrees": 48.94518622903756, "source_dataset": "advanced_geometry", "source_index": 514, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, -7), B=(7, 9), and C=(7, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(279.000, -7.000)", "metadata": {"A": ["6", "-7"], "B": ["7", "9"], "C": ["7", "10"], "ortho": ["279", "-7"], "orthocenter_approx": [279.0, -7.0], "source_dataset": "advanced_geometry", "source_index": 515, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-2, 7), B=(-9, 6), and C=(-10, -5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.336", "metadata": {"A": ["-2", "7"], "B": ["-9", "6"], "C": ["-10", "-5"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + 5*sqrt(2)/2 + 2*sqrt(13))*sqrt(-2*sqrt(13) + 5*sqrt(2)/2 + sqrt(122)/2)*sqrt(-5*sqrt(2)/2 + sqrt(122)/2 + 2*sqrt(13))/sqrt(5*sqrt(2)/2 + sqrt(122)/2 + 2*sqrt(13))", "incircle_radius_approx": 2.3356850248039156, "source_dataset": "advanced_geometry", "source_index": 516, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, 3), B=(0, 1), and C=(6, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.800, -0.600)", "metadata": {"A": ["9", "3"], "B": ["0", "1"], "C": ["6", "-6"], "ortho": ["24/5", "-3/5"], "orthocenter_approx": [4.8, -0.6], "source_dataset": "advanced_geometry", "source_index": 517, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, 5), B=(4, -10), and C=(8, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "59.35°", "metadata": {"A": ["-9", "5"], "B": ["4", "-10"], "C": ["8", "2"], "angle_ABC_degrees": 59.34933204294713, "source_dataset": "advanced_geometry", "source_index": 518, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -4), B=(-5, 7), and C=(-1, -4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "25.02°", "metadata": {"A": ["6", "-4"], "B": ["-5", "7"], "C": ["-1", "-4"], "angle_ABC_degrees": 25.016893478100023, "source_dataset": "advanced_geometry", "source_index": 519, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, -7), B=(-9, -4), and C=(-7, 2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "88.26°", "metadata": {"A": ["1", "-7"], "B": ["-9", "-4"], "C": ["-7", "2"], "angle_ABC_degrees": 88.2642954110716, "source_dataset": "advanced_geometry", "source_index": 520, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 10), B=(-10, -5), and C=(-8, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "12.53°", "metadata": {"A": ["-1", "10"], "B": ["-10", "-5"], "C": ["-8", "1"], "angle_ABC_degrees": 12.528807709151511, "source_dataset": "advanced_geometry", "source_index": 521, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, 0), B=(-9, -1), and C=(3, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.082", "metadata": {"A": ["-4", "0"], "B": ["-9", "-1"], "C": ["3", "1"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(26)/2 + sqrt(37))*sqrt(-sqrt(26)/2 + 5*sqrt(2)/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(26)/2 + 5*sqrt(2)/2)/sqrt(sqrt(26)/2 + 5*sqrt(2)/2 + sqrt(37))", "incircle_radius_approx": 0.08218408348524164, "source_dataset": "advanced_geometry", "source_index": 522, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(5, 0), B=(2, 3), and C=(3, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.57°", "metadata": {"A": ["5", "0"], "B": ["2", "3"], "C": ["3", "0"], "angle_ABC_degrees": 26.56505117707799, "source_dataset": "advanced_geometry", "source_index": 523, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, -1), B=(-6, 3), and C=(-8, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.674", "metadata": {"A": ["7", "-1"], "B": ["-6", "3"], "C": ["-8", "2"], "incircle_radius_exact": "sqrt(-sqrt(185)/2 + sqrt(5)/2 + 3*sqrt(26)/2)*sqrt(-3*sqrt(26)/2 + sqrt(5)/2 + sqrt(185)/2)*sqrt(-sqrt(5)/2 + sqrt(185)/2 + 3*sqrt(26)/2)/sqrt(sqrt(5)/2 + sqrt(185)/2 + 3*sqrt(26)/2)", "incircle_radius_approx": 0.674490823882499, "source_dataset": "advanced_geometry", "source_index": 524, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 5), B=(-8, 8), and C=(-1, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.818, 1.818)", "metadata": {"A": ["0", "5"], "B": ["-8", "8"], "C": ["-1", "4"], "ortho": ["-20/11", "20/11"], "orthocenter_approx": [-1.8181818181818181, 1.8181818181818181], "source_dataset": "advanced_geometry", "source_index": 525, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 7), B=(5, -2), and C=(2, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.200, -6.800)", "metadata": {"A": ["-9", "7"], "B": ["5", "-2"], "C": ["2", "-4"], "ortho": ["1/5", "-34/5"], "orthocenter_approx": [0.2, -6.8], "source_dataset": "advanced_geometry", "source_index": 526, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, -9), B=(0, -9), and C=(0, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "90.00°", "metadata": {"A": ["7", "-9"], "B": ["0", "-9"], "C": ["0", "-2"], "angle_ABC_degrees": 90.0, "source_dataset": "advanced_geometry", "source_index": 527, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -10), B=(0, -9), and C=(-10, -2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.947, -14.211)", "metadata": {"A": ["-4", "-10"], "B": ["0", "-9"], "C": ["-10", "-2"], "ortho": ["-132/19", "-270/19"], "orthocenter_approx": [-6.947368421052632, -14.210526315789474], "source_dataset": "advanced_geometry", "source_index": 528, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-2, -7), B=(9, 3), and C=(-9, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.444, -21.889)", "metadata": {"A": ["-2", "-7"], "B": ["9", "3"], "C": ["-9", "-6"], "ortho": ["49/9", "-197/9"], "orthocenter_approx": [5.444444444444445, -21.88888888888889], "source_dataset": "advanced_geometry", "source_index": 529, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, -2), B=(-2, 9), and C=(-2, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-29.500, -2.000)", "metadata": {"A": ["2", "-2"], "B": ["-2", "9"], "C": ["-2", "8"], "ortho": ["-59/2", "-2"], "orthocenter_approx": [-29.5, -2.0], "source_dataset": "advanced_geometry", "source_index": 530, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -2), B=(-3, -8), and C=(2, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "46.79°", "metadata": {"A": ["10", "-2"], "B": ["-3", "-8"], "C": ["2", "7"], "angle_ABC_degrees": 46.78991060824606, "source_dataset": "advanced_geometry", "source_index": 531, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 7), B=(-7, -1), and C=(-3, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.815", "metadata": {"A": ["8", "7"], "B": ["-7", "-1"], "C": ["-3", "3"], "incircle_radius_exact": "sqrt(-17/2 + 2*sqrt(2) + sqrt(137)/2)*sqrt(-sqrt(137)/2 + 2*sqrt(2) + 17/2)*sqrt(-2*sqrt(2) + sqrt(137)/2 + 17/2)/sqrt(2*sqrt(2) + sqrt(137)/2 + 17/2)", "incircle_radius_approx": 0.8148641900610483, "source_dataset": "advanced_geometry", "source_index": 532, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 2), B=(-7, 3), and C=(5, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.424", "metadata": {"A": ["-9", "2"], "B": ["-7", "3"], "C": ["5", "3"], "incircle_radius_exact": "sqrt(-6 + sqrt(5)/2 + sqrt(197)/2)*sqrt(-sqrt(197)/2 + sqrt(5)/2 + 6)*sqrt(-sqrt(5)/2 + 6 + sqrt(197)/2)/sqrt(sqrt(5)/2 + 6 + sqrt(197)/2)", "incircle_radius_approx": 0.42445216840511246, "source_dataset": "advanced_geometry", "source_index": 533, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, 5), B=(-4, -10), and C=(8, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "11.53°", "metadata": {"A": ["8", "5"], "B": ["-4", "-10"], "C": ["8", "0"], "angle_ABC_degrees": 11.534620653644716, "source_dataset": "advanced_geometry", "source_index": 534, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, -8), B=(0, 4), and C=(3, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.974", "metadata": {"A": ["1", "-8"], "B": ["0", "4"], "C": ["3", "-7"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + sqrt(5)/2 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + sqrt(5)/2 + sqrt(130)/2)*sqrt(-sqrt(5)/2 + sqrt(130)/2 + sqrt(145)/2)/sqrt(sqrt(5)/2 + sqrt(130)/2 + sqrt(145)/2)", "incircle_radius_approx": 0.973542358364978, "source_dataset": "advanced_geometry", "source_index": 535, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 8), B=(2, 1), and C=(8, 1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.370", "metadata": {"A": ["-5", "8"], "B": ["2", "1"], "C": ["8", "1"], "incircle_radius_exact": "sqrt(-3 + 7*sqrt(2)/2 + sqrt(218)/2)*sqrt(-7*sqrt(2)/2 + 3 + sqrt(218)/2)*sqrt(-sqrt(218)/2 + 3 + 7*sqrt(2)/2)/sqrt(3 + 7*sqrt(2)/2 + sqrt(218)/2)", "incircle_radius_approx": 1.3696701163978673, "source_dataset": "advanced_geometry", "source_index": 536, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 5), B=(-6, 10), and C=(0, -9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.805", "metadata": {"A": ["2", "5"], "B": ["-6", "10"], "C": ["0", "-9"], "incircle_radius_exact": "sqrt(-5*sqrt(2) + sqrt(89)/2 + sqrt(397)/2)*sqrt(-sqrt(397)/2 + sqrt(89)/2 + 5*sqrt(2))*sqrt(-sqrt(89)/2 + 5*sqrt(2) + sqrt(397)/2)/sqrt(sqrt(89)/2 + 5*sqrt(2) + sqrt(397)/2)", "incircle_radius_approx": 2.8045348021415624, "source_dataset": "advanced_geometry", "source_index": 537, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -1), B=(-8, 10), and C=(-1, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "38.16°", "metadata": {"A": ["6", "-1"], "B": ["-8", "10"], "C": ["-1", "10"], "angle_ABC_degrees": 38.15722658736906, "source_dataset": "advanced_geometry", "source_index": 538, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, -5), B=(-1, -1), and C=(2, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "84.47°", "metadata": {"A": ["-10", "-5"], "B": ["-1", "-1"], "C": ["2", "-10"], "angle_ABC_degrees": 84.47245984834382, "source_dataset": "advanced_geometry", "source_index": 539, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, -3), B=(4, -5), and C=(2, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.250, -7.625)", "metadata": {"A": ["-7", "-3"], "B": ["4", "-5"], "C": ["2", "-9"], "ortho": ["9/4", "-61/8"], "orthocenter_approx": [2.25, -7.625], "source_dataset": "advanced_geometry", "source_index": 540, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -1), B=(-6, 8), and C=(10, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "99.92°", "metadata": {"A": ["-7", "-1"], "B": ["-6", "8"], "C": ["10", "9"], "angle_ABC_degrees": 99.91652612090726, "source_dataset": "advanced_geometry", "source_index": 541, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, -6), B=(-8, 5), and C=(-1, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "91.13°", "metadata": {"A": ["-2", "-6"], "B": ["-8", "5"], "C": ["-1", "9"], "angle_ABC_degrees": 91.134421630977, "source_dataset": "advanced_geometry", "source_index": 542, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, -4), B=(4, 9), and C=(1, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(16.281, -13.105)", "metadata": {"A": ["-8", "-4"], "B": ["4", "9"], "C": ["1", "1"], "ortho": ["928/57", "-249/19"], "orthocenter_approx": [16.280701754385966, -13.105263157894736], "source_dataset": "advanced_geometry", "source_index": 543, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, -7), B=(9, 1), and C=(-8, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "49.27°", "metadata": {"A": ["-2", "-7"], "B": ["9", "1"], "C": ["-8", "5"], "angle_ABC_degrees": 49.26789330029081, "source_dataset": "advanced_geometry", "source_index": 544, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(0, -9), B=(10, 7), and C=(6, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.68°", "metadata": {"A": ["0", "-9"], "B": ["10", "7"], "C": ["6", "1"], "angle_ABC_degrees": 1.6846843178962914, "source_dataset": "advanced_geometry", "source_index": 545, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -1), B=(-3, -10), and C=(0, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.05°", "metadata": {"A": ["2", "-1"], "B": ["-3", "-10"], "C": ["0", "7"], "angle_ABC_degrees": 19.04662429763581, "source_dataset": "advanced_geometry", "source_index": 546, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 10), B=(-3, -1), and C=(10, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.299, 4.373)", "metadata": {"A": ["0", "10"], "B": ["-3", "-1"], "C": ["10", "2"], "ortho": ["87/67", "293/67"], "orthocenter_approx": [1.2985074626865671, 4.3731343283582085], "source_dataset": "advanced_geometry", "source_index": 547, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 7), B=(5, 2), and C=(3, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.750, -9.500)", "metadata": {"A": ["-9", "7"], "B": ["5", "2"], "C": ["3", "1"], "ortho": ["-3/4", "-19/2"], "orthocenter_approx": [-0.75, -9.5], "source_dataset": "advanced_geometry", "source_index": 548, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, 6), B=(-7, 4), and C=(-6, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "97.77°", "metadata": {"A": ["2", "6"], "B": ["-7", "4"], "C": ["-6", "-8"], "angle_ABC_degrees": 97.76516601842535, "source_dataset": "advanced_geometry", "source_index": 549, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 6), B=(-5, -8), and C=(-6, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-14.400, 1.400)", "metadata": {"A": ["4", "6"], "B": ["-5", "-8"], "C": ["-6", "-4"], "ortho": ["-72/5", "7/5"], "orthocenter_approx": [-14.4, 1.4], "source_dataset": "advanced_geometry", "source_index": 550, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 1), B=(-8, 5), and C=(2, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.987", "metadata": {"A": ["1", "1"], "B": ["-8", "5"], "C": ["2", "3"], "incircle_radius_exact": "sqrt(-sqrt(26) + sqrt(5)/2 + sqrt(97)/2)*sqrt(-sqrt(5)/2 + sqrt(97)/2 + sqrt(26))*sqrt(-sqrt(97)/2 + sqrt(5)/2 + sqrt(26))/sqrt(sqrt(5)/2 + sqrt(97)/2 + sqrt(26))", "incircle_radius_approx": 0.9873012945566759, "source_dataset": "advanced_geometry", "source_index": 551, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 9), B=(-4, -1), and C=(-9, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-106.273, 78.545)", "metadata": {"A": ["5", "9"], "B": ["-4", "-1"], "C": ["-9", "-9"], "ortho": ["-1169/11", "864/11"], "orthocenter_approx": [-106.27272727272727, 78.54545454545455], "source_dataset": "advanced_geometry", "source_index": 552, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, 10), B=(6, 3), and C=(-10, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.324", "metadata": {"A": ["1", "10"], "B": ["6", "3"], "C": ["-10", "-3"], "incircle_radius_exact": "sqrt(-sqrt(290)/2 + sqrt(74)/2 + sqrt(73))*sqrt(-sqrt(73) + sqrt(74)/2 + sqrt(290)/2)*sqrt(-sqrt(74)/2 + sqrt(290)/2 + sqrt(73))/sqrt(sqrt(74)/2 + sqrt(290)/2 + sqrt(73))", "incircle_radius_approx": 3.323991892108209, "source_dataset": "advanced_geometry", "source_index": 553, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, 4), B=(9, -10), and C=(5, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.005", "metadata": {"A": ["0", "4"], "B": ["9", "-10"], "C": ["5", "0"], "incircle_radius_exact": "sqrt(-sqrt(277)/2 + sqrt(41)/2 + sqrt(29))*sqrt(-sqrt(29) + sqrt(41)/2 + sqrt(277)/2)*sqrt(-sqrt(41)/2 + sqrt(29) + sqrt(277)/2)/sqrt(sqrt(41)/2 + sqrt(29) + sqrt(277)/2)", "incircle_radius_approx": 1.0054182929568451, "source_dataset": "advanced_geometry", "source_index": 554, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -2), B=(3, -7), and C=(7, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.644, -5.842)", "metadata": {"A": ["-6", "-2"], "B": ["3", "-7"], "C": ["7", "2"], "ortho": ["267/101", "-590/101"], "orthocenter_approx": [2.6435643564356437, -5.841584158415841], "source_dataset": "advanced_geometry", "source_index": 555, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 6), B=(-7, 4), and C=(1, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.367, -1.020)", "metadata": {"A": ["4", "6"], "B": ["-7", "4"], "C": ["1", "1"], "ortho": ["67/49", "-50/49"], "orthocenter_approx": [1.3673469387755102, -1.0204081632653061], "source_dataset": "advanced_geometry", "source_index": 556, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, 0), B=(-7, 8), and C=(3, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "44.50°", "metadata": {"A": ["-9", "0"], "B": ["-7", "8"], "C": ["3", "-9"], "angle_ABC_degrees": 44.50178838738636, "source_dataset": "advanced_geometry", "source_index": 557, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, 10), B=(4, -6), and C=(9, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.917", "metadata": {"A": ["1", "10"], "B": ["4", "-6"], "C": ["9", "-7"], "incircle_radius_exact": "sqrt(-sqrt(26)/2 + sqrt(265)/2 + sqrt(353)/2)*sqrt(-sqrt(265)/2 + sqrt(26)/2 + sqrt(353)/2)*sqrt(-sqrt(353)/2 + sqrt(26)/2 + sqrt(265)/2)/sqrt(sqrt(26)/2 + sqrt(265)/2 + sqrt(353)/2)", "incircle_radius_approx": 1.91703785463952, "source_dataset": "advanced_geometry", "source_index": 558, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, 7), B=(-9, -7), and C=(4, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(50.828, -54.862)", "metadata": {"A": ["8", "7"], "B": ["-9", "-7"], "C": ["4", "2"], "ortho": ["1474/29", "-1591/29"], "orthocenter_approx": [50.827586206896555, -54.86206896551724], "source_dataset": "advanced_geometry", "source_index": 559, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 0), B=(6, -1), and C=(4, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.179", "metadata": {"A": ["-5", "0"], "B": ["6", "-1"], "C": ["4", "9"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + sqrt(26) + 9*sqrt(2)/2)*sqrt(-9*sqrt(2)/2 + sqrt(26) + sqrt(122)/2)*sqrt(-sqrt(26) + sqrt(122)/2 + 9*sqrt(2)/2)/sqrt(sqrt(26) + sqrt(122)/2 + 9*sqrt(2)/2)", "incircle_radius_approx": 3.179152099640575, "source_dataset": "advanced_geometry", "source_index": 560, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(7, 8), B=(0, 4), and C=(-10, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.000, -9.500)", "metadata": {"A": ["7", "8"], "B": ["0", "4"], "C": ["-10", "8"], "ortho": ["0", "-19/2"], "orthocenter_approx": [0.0, -9.5], "source_dataset": "advanced_geometry", "source_index": 561, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, 2), B=(1, 8), and C=(1, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, 2.000)", "metadata": {"A": ["-4", "2"], "B": ["1", "8"], "C": ["1", "2"], "ortho": ["1", "2"], "orthocenter_approx": [1.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 562, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, 3), B=(7, 5), and C=(-7, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(11.000, 0.000)", "metadata": {"A": ["8", "3"], "B": ["7", "5"], "C": ["-7", "-9"], "ortho": ["11", "0"], "orthocenter_approx": [11.0, 0.0], "source_dataset": "advanced_geometry", "source_index": 563, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, 1), B=(3, 1), and C=(-6, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.43°", "metadata": {"A": ["-9", "1"], "B": ["3", "1"], "C": ["-6", "4"], "angle_ABC_degrees": 18.43494882292201, "source_dataset": "advanced_geometry", "source_index": 564, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, -7), B=(-6, 0), and C=(-9, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "15.02°", "metadata": {"A": ["-8", "-7"], "B": ["-6", "0"], "C": ["-9", "-5"], "angle_ABC_degrees": 15.018360631150667, "source_dataset": "advanced_geometry", "source_index": 565, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 5), B=(9, 4), and C=(7, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.756, -2.171)", "metadata": {"A": ["-4", "5"], "B": ["9", "4"], "C": ["7", "1"], "ortho": ["277/41", "-89/41"], "orthocenter_approx": [6.7560975609756095, -2.1707317073170733], "source_dataset": "advanced_geometry", "source_index": 566, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, -1), B=(-9, -5), and C=(10, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "25.05°", "metadata": {"A": ["8", "-1"], "B": ["-9", "-5"], "C": ["10", "10"], "angle_ABC_degrees": 25.049643277055864, "source_dataset": "advanced_geometry", "source_index": 567, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(3, -1), B=(-1, 1), and C=(0, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.303", "metadata": {"A": ["3", "-1"], "B": ["-1", "1"], "C": ["0", "-4"], "incircle_radius_exact": "sqrt(-3*sqrt(2)/2 + sqrt(5) + sqrt(26)/2)*sqrt(-sqrt(26)/2 + 3*sqrt(2)/2 + sqrt(5))*sqrt(-sqrt(5) + 3*sqrt(2)/2 + sqrt(26)/2)/sqrt(3*sqrt(2)/2 + sqrt(5) + sqrt(26)/2)", "incircle_radius_approx": 1.3030451439344182, "source_dataset": "advanced_geometry", "source_index": 568, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, 10), B=(-6, 10), and C=(0, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.57°", "metadata": {"A": ["1", "10"], "B": ["-6", "10"], "C": ["0", "7"], "angle_ABC_degrees": 26.56505117707799, "source_dataset": "advanced_geometry", "source_index": 569, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 7), B=(9, 8), and C=(8, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.310, 5.972)", "metadata": {"A": ["-4", "7"], "B": ["9", "8"], "C": ["8", "-3"], "ortho": ["519/71", "424/71"], "orthocenter_approx": [7.309859154929577, 5.971830985915493], "source_dataset": "advanced_geometry", "source_index": 570, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, -7), B=(6, 4), and C=(4, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.996", "metadata": {"A": ["-6", "-7"], "B": ["6", "4"], "C": ["4", "5"], "incircle_radius_exact": "sqrt(-sqrt(265)/2 + sqrt(5)/2 + sqrt(61))*sqrt(-sqrt(5)/2 + sqrt(61) + sqrt(265)/2)*sqrt(-sqrt(61) + sqrt(5)/2 + sqrt(265)/2)/sqrt(sqrt(5)/2 + sqrt(61) + sqrt(265)/2)", "incircle_radius_approx": 0.9960337956109175, "source_dataset": "advanced_geometry", "source_index": 571, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 9), B=(-4, -8), and C=(8, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.187", "metadata": {"A": ["-3", "9"], "B": ["-4", "-8"], "C": ["8", "-7"], "incircle_radius_exact": "sqrt(-sqrt(145)/2 + sqrt(290)/2 + sqrt(377)/2)*sqrt(-sqrt(290)/2 + sqrt(145)/2 + sqrt(377)/2)*sqrt(-sqrt(377)/2 + sqrt(145)/2 + sqrt(290)/2)/sqrt(sqrt(145)/2 + sqrt(290)/2 + sqrt(377)/2)", "incircle_radius_approx": 4.186648738166004, "source_dataset": "advanced_geometry", "source_index": 572, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, -8), B=(-5, -6), and C=(-1, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.440, -5.040)", "metadata": {"A": ["2", "-8"], "B": ["-5", "-6"], "C": ["-1", "0"], "ortho": ["-61/25", "-126/25"], "orthocenter_approx": [-2.44, -5.04], "source_dataset": "advanced_geometry", "source_index": 573, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, 6), B=(7, -5), and C=(0, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.000, 7.909)", "metadata": {"A": ["4", "6"], "B": ["7", "-5"], "C": ["0", "6"], "ortho": ["7", "87/11"], "orthocenter_approx": [7.0, 7.909090909090909], "source_dataset": "advanced_geometry", "source_index": 574, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 4), B=(-2, 0), and C=(-8, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "91.22°", "metadata": {"A": ["5", "4"], "B": ["-2", "0"], "C": ["-8", "10"], "angle_ABC_degrees": 91.2188752351313, "source_dataset": "advanced_geometry", "source_index": 575, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 4), B=(-7, 0), and C=(-5, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.971", "metadata": {"A": ["-5", "4"], "B": ["-7", "0"], "C": ["-5", "-8"], "incircle_radius_exact": "sqrt(-6 + sqrt(5) + sqrt(17))*sqrt(-sqrt(17) + sqrt(5) + 6)*sqrt(-sqrt(5) + sqrt(17) + 6)/sqrt(sqrt(5) + sqrt(17) + 6)", "incircle_radius_approx": 0.9709387039415925, "source_dataset": "advanced_geometry", "source_index": 576, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, -6), B=(5, 10), and C=(-5, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.15°", "metadata": {"A": ["-10", "-6"], "B": ["5", "10"], "C": ["-5", "7"], "angle_ABC_degrees": 30.148366032000972, "source_dataset": "advanced_geometry", "source_index": 577, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, 6), B=(5, 9), and C=(8, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.651, 3.767)", "metadata": {"A": ["3", "6"], "B": ["5", "9"], "C": ["8", "-8"], "ortho": ["-415/43", "162/43"], "orthocenter_approx": [-9.651162790697674, 3.7674418604651163], "source_dataset": "advanced_geometry", "source_index": 578, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 9), B=(6, 1), and C=(1, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.772", "metadata": {"A": ["9", "9"], "B": ["6", "1"], "C": ["1", "2"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + sqrt(26)/2 + sqrt(73)/2)*sqrt(-sqrt(26)/2 + sqrt(73)/2 + sqrt(113)/2)*sqrt(-sqrt(73)/2 + sqrt(26)/2 + sqrt(113)/2)/sqrt(sqrt(26)/2 + sqrt(73)/2 + sqrt(113)/2)", "incircle_radius_approx": 1.7715033365886714, "source_dataset": "advanced_geometry", "source_index": 579, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 5), B=(10, -1), and C=(-10, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.508", "metadata": {"A": ["-9", "5"], "B": ["10", "-1"], "C": ["-10", "9"], "incircle_radius_exact": "sqrt(-sqrt(17)/2 + sqrt(397)/2 + 5*sqrt(5))*sqrt(-sqrt(397)/2 + sqrt(17)/2 + 5*sqrt(5))*sqrt(-5*sqrt(5) + sqrt(17)/2 + sqrt(397)/2)/sqrt(sqrt(17)/2 + sqrt(397)/2 + 5*sqrt(5))", "incircle_radius_approx": 1.5083396883836977, "source_dataset": "advanced_geometry", "source_index": 580, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, -5), B=(-8, 2), and C=(8, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-482.000, -709.000)", "metadata": {"A": ["2", "-5"], "B": ["-8", "2"], "C": ["8", "-9"], "ortho": ["-482", "-709"], "orthocenter_approx": [-482.0, -709.0], "source_dataset": "advanced_geometry", "source_index": 581, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, -3), B=(-4, -1), and C=(8, -2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.000, 33.000)", "metadata": {"A": ["-9", "-3"], "B": ["-4", "-1"], "C": ["8", "-2"], "ortho": ["-6", "33"], "orthocenter_approx": [-6.0, 33.0], "source_dataset": "advanced_geometry", "source_index": 582, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-2, 9), B=(9, -8), and C=(-2, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.828", "metadata": {"A": ["-2", "9"], "B": ["9", "-8"], "C": ["-2", "2"], "incircle_radius_exact": "sqrt(-7/2 + sqrt(221)/2 + sqrt(410)/2)*sqrt(-sqrt(221)/2 + 7/2 + sqrt(410)/2)*sqrt(-sqrt(410)/2 + 7/2 + sqrt(221)/2)/sqrt(7/2 + sqrt(221)/2 + sqrt(410)/2)", "incircle_radius_approx": 1.8283477998360087, "source_dataset": "advanced_geometry", "source_index": 583, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -3), B=(-10, -6), and C=(10, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.991", "metadata": {"A": ["-7", "-3"], "B": ["-10", "-6"], "C": ["10", "0"], "incircle_radius_exact": "sqrt(-sqrt(109) + 3*sqrt(2)/2 + sqrt(298)/2)*sqrt(-3*sqrt(2)/2 + sqrt(298)/2 + sqrt(109))*sqrt(-sqrt(298)/2 + 3*sqrt(2)/2 + sqrt(109))/sqrt(3*sqrt(2)/2 + sqrt(298)/2 + sqrt(109))", "incircle_radius_approx": 0.9908948510816777, "source_dataset": "advanced_geometry", "source_index": 584, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(8, -7), B=(-6, -3), and C=(-10, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.432", "metadata": {"A": ["8", "-7"], "B": ["-6", "-3"], "C": ["-10", "-3"], "incircle_radius_exact": "sqrt(-2 + sqrt(53) + sqrt(85))*sqrt(-sqrt(53) + 2 + sqrt(85))*sqrt(-sqrt(85) + 2 + sqrt(53))/sqrt(2 + sqrt(53) + sqrt(85))", "incircle_radius_approx": 0.43244051213755774, "source_dataset": "advanced_geometry", "source_index": 585, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, 7), B=(7, -7), and C=(4, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.653", "metadata": {"A": ["-8", "7"], "B": ["7", "-7"], "C": ["4", "-6"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(313)/2 + sqrt(421)/2)*sqrt(-sqrt(313)/2 + sqrt(10)/2 + sqrt(421)/2)*sqrt(-sqrt(421)/2 + sqrt(10)/2 + sqrt(313)/2)/sqrt(sqrt(10)/2 + sqrt(313)/2 + sqrt(421)/2)", "incircle_radius_approx": 0.6526094872862893, "source_dataset": "advanced_geometry", "source_index": 586, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, 5), B=(8, -10), and C=(9, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(16.759, 3.138)", "metadata": {"A": ["0", "5"], "B": ["8", "-10"], "C": ["9", "-1"], "ortho": ["486/29", "91/29"], "orthocenter_approx": [16.75862068965517, 3.1379310344827585], "source_dataset": "advanced_geometry", "source_index": 587, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, 6), B=(-8, 9), and C=(1, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.032", "metadata": {"A": ["-6", "6"], "B": ["-8", "9"], "C": ["1", "-4"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(13)/2 + sqrt(149)/2)*sqrt(-sqrt(13)/2 + sqrt(149)/2 + 5*sqrt(10)/2)*sqrt(-sqrt(149)/2 + sqrt(13)/2 + 5*sqrt(10)/2)/sqrt(sqrt(13)/2 + sqrt(149)/2 + 5*sqrt(10)/2)", "incircle_radius_approx": 0.031622058027656746, "source_dataset": "advanced_geometry", "source_index": 588, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, -4), B=(-5, 9), and C=(-6, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.445", "metadata": {"A": ["0", "-4"], "B": ["-5", "9"], "C": ["-6", "9"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(194)/2 + sqrt(205)/2)*sqrt(-sqrt(194)/2 + 1/2 + sqrt(205)/2)*sqrt(-sqrt(205)/2 + 1/2 + sqrt(194)/2)/sqrt(1/2 + sqrt(194)/2 + sqrt(205)/2)", "incircle_radius_approx": 0.4445020497755665, "source_dataset": "advanced_geometry", "source_index": 589, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 7), B=(-9, -10), and C=(-1, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.530", "metadata": {"A": ["-3", "7"], "B": ["-9", "-10"], "C": ["-1", "9"], "incircle_radius_exact": "sqrt(-5*sqrt(13)/2 + sqrt(2) + 5*sqrt(17)/2)*sqrt(-5*sqrt(17)/2 + sqrt(2) + 5*sqrt(13)/2)*sqrt(-sqrt(2) + 5*sqrt(13)/2 + 5*sqrt(17)/2)/sqrt(sqrt(2) + 5*sqrt(13)/2 + 5*sqrt(17)/2)", "incircle_radius_approx": 0.5304820836959044, "source_dataset": "advanced_geometry", "source_index": 590, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, -9), B=(1, -10), and C=(-8, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, -10.000)", "metadata": {"A": ["1", "-9"], "B": ["1", "-10"], "C": ["-8", "-10"], "ortho": ["1", "-10"], "orthocenter_approx": [1.0, -10.0], "source_dataset": "advanced_geometry", "source_index": 591, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, 10), B=(-10, 8), and C=(4, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.048, 23.667)", "metadata": {"A": ["-3", "10"], "B": ["-10", "8"], "C": ["4", "6"], "ortho": ["-22/21", "71/3"], "orthocenter_approx": [-1.0476190476190477, 23.666666666666668], "source_dataset": "advanced_geometry", "source_index": 592, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 8), B=(4, -2), and C=(4, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.550", "metadata": {"A": ["1", "8"], "B": ["4", "-2"], "C": ["4", "-8"], "incircle_radius_exact": "sqrt(-3 + sqrt(109)/2 + sqrt(265)/2)*sqrt(-sqrt(109)/2 + 3 + sqrt(265)/2)*sqrt(-sqrt(265)/2 + 3 + sqrt(109)/2)/sqrt(3 + sqrt(109)/2 + sqrt(265)/2)", "incircle_radius_approx": 0.5501369257874753, "source_dataset": "advanced_geometry", "source_index": 593, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, -5), B=(6, 6), and C=(2, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(200.000, -285.000)", "metadata": {"A": ["-10", "-5"], "B": ["6", "6"], "C": ["2", "3"], "ortho": ["200", "-285"], "orthocenter_approx": [200.0, -285.0], "source_dataset": "advanced_geometry", "source_index": 594, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, -2), B=(-6, -9), and C=(4, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.544", "metadata": {"A": ["-4", "-2"], "B": ["-6", "-9"], "C": ["4", "2"], "incircle_radius_exact": "sqrt(-sqrt(221)/2 + sqrt(53)/2 + 2*sqrt(5))*sqrt(-2*sqrt(5) + sqrt(53)/2 + sqrt(221)/2)*sqrt(-sqrt(53)/2 + 2*sqrt(5) + sqrt(221)/2)/sqrt(sqrt(53)/2 + 2*sqrt(5) + sqrt(221)/2)", "incircle_radius_approx": 1.5438824190745608, "source_dataset": "advanced_geometry", "source_index": 595, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, 5), B=(1, 2), and C=(-8, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "78.50°", "metadata": {"A": ["-3", "5"], "B": ["1", "2"], "C": ["-8", "-6"], "angle_ABC_degrees": 78.50343698241421, "source_dataset": "advanced_geometry", "source_index": 596, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 2), B=(10, -6), and C=(1, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "106.84°", "metadata": {"A": ["9", "2"], "B": ["10", "-6"], "C": ["1", "-10"], "angle_ABC_degrees": 106.83747262567638, "source_dataset": "advanced_geometry", "source_index": 597, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, -5), B=(-9, 1), and C=(-4, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-11.714, -20.714)", "metadata": {"A": ["4", "-5"], "B": ["-9", "1"], "C": ["-4", "-4"], "ortho": ["-82/7", "-145/7"], "orthocenter_approx": [-11.714285714285714, -20.714285714285715], "source_dataset": "advanced_geometry", "source_index": 598, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 8), B=(-8, 3), and C=(1, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.958", "metadata": {"A": ["-10", "8"], "B": ["-8", "3"], "C": ["1", "6"], "incircle_radius_exact": "sqrt(-3*sqrt(10)/2 + sqrt(29)/2 + 5*sqrt(5)/2)*sqrt(-sqrt(29)/2 + 3*sqrt(10)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + sqrt(29)/2 + 3*sqrt(10)/2)/sqrt(sqrt(29)/2 + 3*sqrt(10)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 1.9575978415429738, "source_dataset": "advanced_geometry", "source_index": 599, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, 3), B=(6, 8), and C=(5, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "54.42°", "metadata": {"A": ["-2", "3"], "B": ["6", "8"], "C": ["5", "-8"], "angle_ABC_degrees": 54.418282416919155, "source_dataset": "advanced_geometry", "source_index": 600, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, 6), B=(0, 6), and C=(-4, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "105.95°", "metadata": {"A": ["2", "6"], "B": ["0", "6"], "C": ["-4", "-8"], "angle_ABC_degrees": 105.94539590092285, "source_dataset": "advanced_geometry", "source_index": 601, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, 3), B=(-7, 9), and C=(4, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.212", "metadata": {"A": ["0", "3"], "B": ["-7", "9"], "C": ["4", "8"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + sqrt(41)/2 + sqrt(85)/2)*sqrt(-sqrt(41)/2 + sqrt(85)/2 + sqrt(122)/2)*sqrt(-sqrt(85)/2 + sqrt(41)/2 + sqrt(122)/2)/sqrt(sqrt(41)/2 + sqrt(85)/2 + sqrt(122)/2)", "incircle_radius_approx": 2.212386915619936, "source_dataset": "advanced_geometry", "source_index": 602, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -6), B=(3, -7), and C=(-1, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.714, 12.286)", "metadata": {"A": ["9", "-6"], "B": ["3", "-7"], "C": ["-1", "-10"], "ortho": ["-33/7", "86/7"], "orthocenter_approx": [-4.714285714285714, 12.285714285714286], "source_dataset": "advanced_geometry", "source_index": 603, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, 0), B=(6, 9), and C=(-4, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "91.22°", "metadata": {"A": ["8", "0"], "B": ["6", "9"], "C": ["-4", "7"], "angle_ABC_degrees": 91.2188752351313, "source_dataset": "advanced_geometry", "source_index": 604, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, 1), B=(10, -8), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "17.59°", "metadata": {"A": ["7", "1"], "B": ["10", "-8"], "C": ["2", "3"], "angle_ABC_degrees": 17.5924245621816, "source_dataset": "advanced_geometry", "source_index": 605, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, 8), B=(0, 7), and C=(-3, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "89.22°", "metadata": {"A": ["-5", "8"], "B": ["0", "7"], "C": ["-3", "-7"], "angle_ABC_degrees": 89.2151753970081, "source_dataset": "advanced_geometry", "source_index": 606, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-3, 1), B=(4, 9), and C=(9, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.668", "metadata": {"A": ["-3", "1"], "B": ["4", "9"], "C": ["9", "-8"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(113)/2 + sqrt(314)/2)*sqrt(-sqrt(113)/2 + 15/2 + sqrt(314)/2)*sqrt(-sqrt(314)/2 + sqrt(113)/2 + 15/2)/sqrt(sqrt(113)/2 + 15/2 + sqrt(314)/2)", "incircle_radius_approx": 3.667803912303366, "source_dataset": "advanced_geometry", "source_index": 607, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 2), B=(4, 4), and C=(5, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.557", "metadata": {"A": ["-5", "2"], "B": ["4", "4"], "C": ["5", "-4"], "incircle_radius_exact": "sqrt(-sqrt(34) + sqrt(65)/2 + sqrt(85)/2)*sqrt(-sqrt(65)/2 + sqrt(85)/2 + sqrt(34))*sqrt(-sqrt(85)/2 + sqrt(65)/2 + sqrt(34))/sqrt(sqrt(65)/2 + sqrt(85)/2 + sqrt(34))", "incircle_radius_approx": 2.5566871088333456, "source_dataset": "advanced_geometry", "source_index": 608, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -6), B=(-9, -7), and C=(-4, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.615, -32.538)", "metadata": {"A": ["8", "-6"], "B": ["-9", "-7"], "C": ["-4", "-9"], "ortho": ["-34/13", "-423/13"], "orthocenter_approx": [-2.6153846153846154, -32.53846153846154], "source_dataset": "advanced_geometry", "source_index": 609, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, 8), B=(6, 5), and C=(1, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(47.600, -8.000)", "metadata": {"A": ["6", "8"], "B": ["6", "5"], "C": ["1", "-8"], "ortho": ["238/5", "-8"], "orthocenter_approx": [47.6, -8.0], "source_dataset": "advanced_geometry", "source_index": 610, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 8), B=(-5, 5), and C=(-2, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "35.13°", "metadata": {"A": ["5", "8"], "B": ["-5", "5"], "C": ["-2", "4"], "angle_ABC_degrees": 35.134193056915635, "source_dataset": "advanced_geometry", "source_index": 611, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, 2), B=(10, 0), and C=(10, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["8", "2"], "B": ["10", "0"], "C": ["10", "10"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 612, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, -1), B=(-10, -10), and C=(-4, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(17.000, -10.000)", "metadata": {"A": ["-4", "-1"], "B": ["-10", "-10"], "C": ["-4", "4"], "ortho": ["17", "-10"], "orthocenter_approx": [17.0, -10.0], "source_dataset": "advanced_geometry", "source_index": 613, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, 8), B=(-7, -9), and C=(7, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "12.09°", "metadata": {"A": ["4", "8"], "B": ["-7", "-9"], "C": ["7", "5"], "angle_ABC_degrees": 12.094757077012101, "source_dataset": "advanced_geometry", "source_index": 614, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, -9), B=(-10, -2), and C=(-7, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "15.07°", "metadata": {"A": ["-9", "-9"], "B": ["-10", "-2"], "C": ["-7", "-9"], "angle_ABC_degrees": 15.06848815949221, "source_dataset": "advanced_geometry", "source_index": 615, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, -8), B=(10, 6), and C=(3, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.282, -9.436)", "metadata": {"A": ["-1", "-8"], "B": ["10", "6"], "C": ["3", "-10"], "ortho": ["89/39", "-368/39"], "orthocenter_approx": [2.282051282051282, -9.435897435897436], "source_dataset": "advanced_geometry", "source_index": 616, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 5), B=(-4, -10), and C=(-5, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "25.35°", "metadata": {"A": ["-1", "5"], "B": ["-4", "-10"], "C": ["-5", "-6"], "angle_ABC_degrees": 25.346175941946694, "source_dataset": "advanced_geometry", "source_index": 617, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, 2), B=(-2, -8), and C=(-5, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.370, -12.444)", "metadata": {"A": ["10", "2"], "B": ["-2", "-8"], "C": ["-5", "-6"], "ortho": ["10/27", "-112/9"], "orthocenter_approx": [0.37037037037037035, -12.444444444444445], "source_dataset": "advanced_geometry", "source_index": 618, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, -1), B=(-8, 7), and C=(-5, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.679, -6.434)", "metadata": {"A": ["3", "-1"], "B": ["-8", "7"], "C": ["-5", "0"], "ortho": ["-513/53", "-341/53"], "orthocenter_approx": [-9.679245283018869, -6.433962264150943], "source_dataset": "advanced_geometry", "source_index": 619, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -10), B=(-3, -1), and C=(-2, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "68.96°", "metadata": {"A": ["-7", "-10"], "B": ["-3", "-1"], "C": ["-2", "-2"], "angle_ABC_degrees": 68.96248897457818, "source_dataset": "advanced_geometry", "source_index": 620, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -7), B=(4, -7), and C=(10, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "65.22°", "metadata": {"A": ["9", "-7"], "B": ["4", "-7"], "C": ["10", "6"], "angle_ABC_degrees": 65.22485943116808, "source_dataset": "advanced_geometry", "source_index": 621, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 5), B=(10, 8), and C=(-6, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.52°", "metadata": {"A": ["6", "5"], "B": ["10", "8"], "C": ["-6", "3"], "angle_ABC_degrees": 19.515873009582698, "source_dataset": "advanced_geometry", "source_index": 622, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -3), B=(6, 4), and C=(1, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "58.74°", "metadata": {"A": ["9", "-3"], "B": ["6", "4"], "C": ["1", "-3"], "angle_ABC_degrees": 58.73626830562257, "source_dataset": "advanced_geometry", "source_index": 623, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 3), B=(-1, 6), and C=(-10, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.925", "metadata": {"A": ["2", "3"], "B": ["-1", "6"], "C": ["-10", "7"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + 3*sqrt(2)/2 + sqrt(82)/2)*sqrt(-3*sqrt(2)/2 + sqrt(82)/2 + 2*sqrt(10))*sqrt(-sqrt(82)/2 + 3*sqrt(2)/2 + 2*sqrt(10))/sqrt(3*sqrt(2)/2 + sqrt(82)/2 + 2*sqrt(10))", "incircle_radius_approx": 0.9249575586698402, "source_dataset": "advanced_geometry", "source_index": 624, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(7, 3), B=(3, 1), and C=(-1, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.000, -15.000)", "metadata": {"A": ["7", "3"], "B": ["3", "1"], "C": ["-1", "1"], "ortho": ["7", "-15"], "orthocenter_approx": [7.0, -15.0], "source_dataset": "advanced_geometry", "source_index": 625, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-7, -10), B=(3, -5), and C=(-9, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.100", "metadata": {"A": ["-7", "-10"], "B": ["3", "-5"], "C": ["-9", "-5"], "incircle_radius_exact": "sqrt(-6 + sqrt(29)/2 + 5*sqrt(5)/2)*sqrt(-sqrt(29)/2 + 5*sqrt(5)/2 + 6)*sqrt(-5*sqrt(5)/2 + sqrt(29)/2 + 6)/sqrt(sqrt(29)/2 + 5*sqrt(5)/2 + 6)", "incircle_radius_approx": 2.100435495238146, "source_dataset": "advanced_geometry", "source_index": 626, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 6), B=(5, 10), and C=(-4, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.310", "metadata": {"A": ["9", "6"], "B": ["5", "10"], "C": ["-4", "-1"], "incircle_radius_exact": "sqrt(-2*sqrt(2) + sqrt(202)/2 + sqrt(218)/2)*sqrt(-sqrt(202)/2 + 2*sqrt(2) + sqrt(218)/2)*sqrt(-sqrt(218)/2 + 2*sqrt(2) + sqrt(202)/2)/sqrt(2*sqrt(2) + sqrt(202)/2 + sqrt(218)/2)", "incircle_radius_approx": 2.309845724893806, "source_dataset": "advanced_geometry", "source_index": 627, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, -4), B=(3, -6), and C=(2, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "37.87°", "metadata": {"A": ["1", "-4"], "B": ["3", "-6"], "C": ["2", "2"], "angle_ABC_degrees": 37.874983651098205, "source_dataset": "advanced_geometry", "source_index": 628, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, 2), B=(4, 2), and C=(0, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "71.57°", "metadata": {"A": ["-6", "2"], "B": ["4", "2"], "C": ["0", "-10"], "angle_ABC_degrees": 71.56505117707799, "source_dataset": "advanced_geometry", "source_index": 629, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, -4), B=(10, -5), and C=(7, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.477, -0.890)", "metadata": {"A": ["-7", "-4"], "B": ["10", "-5"], "C": ["7", "8"], "ortho": ["706/109", "-97/109"], "orthocenter_approx": [6.477064220183486, -0.8899082568807339], "source_dataset": "advanced_geometry", "source_index": 630, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 4), B=(2, -5), and C=(-10, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "57.09°", "metadata": {"A": ["1", "4"], "B": ["2", "-5"], "C": ["-10", "1"], "angle_ABC_degrees": 57.094757077012105, "source_dataset": "advanced_geometry", "source_index": 631, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, -8), B=(-6, 10), and C=(-5, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.088", "metadata": {"A": ["-8", "-8"], "B": ["-6", "10"], "C": ["-5", "-1"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + sqrt(58)/2 + sqrt(82))*sqrt(-sqrt(58)/2 + sqrt(122)/2 + sqrt(82))*sqrt(-sqrt(82) + sqrt(58)/2 + sqrt(122)/2)/sqrt(sqrt(58)/2 + sqrt(122)/2 + sqrt(82))", "incircle_radius_approx": 1.0877870116711488, "source_dataset": "advanced_geometry", "source_index": 632, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, 2), B=(-1, 10), and C=(5, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-12.538, -5.385)", "metadata": {"A": ["1", "2"], "B": ["-1", "10"], "C": ["5", "-1"], "ortho": ["-163/13", "-70/13"], "orthocenter_approx": [-12.538461538461538, -5.384615384615385], "source_dataset": "advanced_geometry", "source_index": 633, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 5), B=(7, 7), and C=(3, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(8.364, 5.364)", "metadata": {"A": ["9", "5"], "B": ["7", "7"], "C": ["3", "0"], "ortho": ["92/11", "59/11"], "orthocenter_approx": [8.363636363636363, 5.363636363636363], "source_dataset": "advanced_geometry", "source_index": 634, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-8, 10), B=(-8, -6), and C=(5, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "42.88°", "metadata": {"A": ["-8", "10"], "B": ["-8", "-6"], "C": ["5", "8"], "angle_ABC_degrees": 42.87890360333855, "source_dataset": "advanced_geometry", "source_index": 635, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, -10), B=(-6, 8), and C=(-7, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-101.000, 20.667)", "metadata": {"A": ["-9", "-10"], "B": ["-6", "8"], "C": ["-7", "5"], "ortho": ["-101", "62/3"], "orthocenter_approx": [-101.0, 20.666666666666668], "source_dataset": "advanced_geometry", "source_index": 636, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, 7), B=(2, 1), and C=(-8, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "112.83°", "metadata": {"A": ["9", "7"], "B": ["2", "1"], "C": ["-8", "6"], "angle_ABC_degrees": 112.83365417791755, "source_dataset": "advanced_geometry", "source_index": 637, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 8), B=(-6, 6), and C=(7, 1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.34°", "metadata": {"A": ["5", "8"], "B": ["-6", "6"], "C": ["7", "1"], "angle_ABC_degrees": 31.342357494187848, "source_dataset": "advanced_geometry", "source_index": 638, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, -7), B=(4, 2), and C=(-9, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.378", "metadata": {"A": ["0", "-7"], "B": ["4", "2"], "C": ["-9", "5"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(97)/2 + sqrt(178)/2)*sqrt(-sqrt(178)/2 + sqrt(97)/2 + 15/2)*sqrt(-sqrt(97)/2 + sqrt(178)/2 + 15/2)/sqrt(sqrt(97)/2 + sqrt(178)/2 + 15/2)", "incircle_radius_approx": 3.37780144646589, "source_dataset": "advanced_geometry", "source_index": 639, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -4), B=(8, -9), and C=(2, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "46.12°", "metadata": {"A": ["-3", "-4"], "B": ["8", "-9"], "C": ["2", "8"], "angle_ABC_degrees": 46.11601039140727, "source_dataset": "advanced_geometry", "source_index": 640, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, -2), B=(-4, -6), and C=(-7, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.830", "metadata": {"A": ["-9", "-2"], "B": ["-4", "-6"], "C": ["-7", "7"], "incircle_radius_exact": "sqrt(-sqrt(178)/2 + sqrt(41)/2 + sqrt(85)/2)*sqrt(-sqrt(41)/2 + sqrt(85)/2 + sqrt(178)/2)*sqrt(-sqrt(85)/2 + sqrt(41)/2 + sqrt(178)/2)/sqrt(sqrt(41)/2 + sqrt(85)/2 + sqrt(178)/2)", "incircle_radius_approx": 1.8298367319993643, "source_dataset": "advanced_geometry", "source_index": 641, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 3), B=(6, 7), and C=(4, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.644", "metadata": {"A": ["9", "3"], "B": ["6", "7"], "C": ["4", "-2"], "incircle_radius_exact": "sqrt(-5/2 + 5*sqrt(2)/2 + sqrt(85)/2)*sqrt(-5*sqrt(2)/2 + 5/2 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + 5/2 + 5*sqrt(2)/2)/sqrt(5/2 + 5*sqrt(2)/2 + sqrt(85)/2)", "incircle_radius_approx": 1.6439170258481055, "source_dataset": "advanced_geometry", "source_index": 642, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, -3), B=(-10, -1), and C=(9, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.474", "metadata": {"A": ["-8", "-3"], "B": ["-10", "-1"], "C": ["9", "-10"], "incircle_radius_exact": "sqrt(-11*sqrt(2)/2 + sqrt(442)/2)*sqrt(11*sqrt(2)/2 + sqrt(442)/2)*sqrt(-sqrt(442)/2 + 15*sqrt(2)/2)/sqrt(sqrt(442)/2 + 15*sqrt(2)/2)", "incircle_radius_approx": 0.47351848491946863, "source_dataset": "advanced_geometry", "source_index": 643, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, -1), B=(-1, -8), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "33.56°", "metadata": {"A": ["7", "-1"], "B": ["-1", "-8"], "C": ["2", "3"], "angle_ABC_degrees": 33.55895613123258, "source_dataset": "advanced_geometry", "source_index": 644, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, 9), B=(5, 10), and C=(-10, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.426", "metadata": {"A": ["-6", "9"], "B": ["5", "10"], "C": ["-10", "-7"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + 2*sqrt(17) + sqrt(514)/2)*sqrt(-2*sqrt(17) + sqrt(122)/2 + sqrt(514)/2)*sqrt(-sqrt(514)/2 + sqrt(122)/2 + 2*sqrt(17))/sqrt(sqrt(122)/2 + 2*sqrt(17) + sqrt(514)/2)", "incircle_radius_approx": 3.425656664747512, "source_dataset": "advanced_geometry", "source_index": 645, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, -4), B=(-8, -10), and C=(4, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.96°", "metadata": {"A": ["-2", "-4"], "B": ["-8", "-10"], "C": ["4", "-7"], "angle_ABC_degrees": 30.96375653207352, "source_dataset": "advanced_geometry", "source_index": 646, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, 0), B=(-10, 5), and C=(8, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "51.34°", "metadata": {"A": ["-5", "0"], "B": ["-10", "5"], "C": ["8", "7"], "angle_ABC_degrees": 51.340191745909905, "source_dataset": "advanced_geometry", "source_index": 647, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, -5), B=(5, -8), and C=(-6, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.436", "metadata": {"A": ["4", "-5"], "B": ["5", "-8"], "C": ["-6", "8"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(269)/2 + sqrt(377)/2)*sqrt(-sqrt(269)/2 + sqrt(10)/2 + sqrt(377)/2)*sqrt(-sqrt(377)/2 + sqrt(10)/2 + sqrt(269)/2)/sqrt(sqrt(10)/2 + sqrt(269)/2 + sqrt(377)/2)", "incircle_radius_approx": 0.43612125594301865, "source_dataset": "advanced_geometry", "source_index": 648, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -6), B=(-7, 4), and C=(-1, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "50.60°", "metadata": {"A": ["-6", "-6"], "B": ["-7", "4"], "C": ["-1", "0"], "angle_ABC_degrees": 50.59933933652058, "source_dataset": "advanced_geometry", "source_index": 649, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, -6), B=(7, 4), and C=(6, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.649, -5.405)", "metadata": {"A": ["10", "-6"], "B": ["7", "4"], "C": ["6", "-5"], "ortho": ["172/37", "-200/37"], "orthocenter_approx": [4.648648648648648, -5.405405405405405], "source_dataset": "advanced_geometry", "source_index": 650, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 8), B=(1, -6), and C=(1, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.432", "metadata": {"A": ["-3", "8"], "B": ["1", "-6"], "C": ["1", "-10"], "incircle_radius_exact": "sqrt(-2 + sqrt(53) + sqrt(85))*sqrt(-sqrt(53) + 2 + sqrt(85))*sqrt(-sqrt(85) + 2 + sqrt(53))/sqrt(2 + sqrt(53) + sqrt(85))", "incircle_radius_approx": 0.43244051213755774, "source_dataset": "advanced_geometry", "source_index": 651, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -9), B=(-9, -2), and C=(1, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "72.15°", "metadata": {"A": ["-1", "-9"], "B": ["-9", "-2"], "C": ["1", "4"], "angle_ABC_degrees": 72.14968169778317, "source_dataset": "advanced_geometry", "source_index": 652, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-3, 7), B=(1, 7), and C=(3, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.498", "metadata": {"A": ["-3", "7"], "B": ["1", "7"], "C": ["3", "-2"], "incircle_radius_exact": "sqrt(-2 + sqrt(85)/2 + 3*sqrt(13)/2)*sqrt(-3*sqrt(13)/2 + 2 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + 2 + 3*sqrt(13)/2)/sqrt(2 + sqrt(85)/2 + 3*sqrt(13)/2)", "incircle_radius_approx": 1.4977410144114123, "source_dataset": "advanced_geometry", "source_index": 653, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-3, 4), B=(10, 9), and C=(5, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.321, 1.366)", "metadata": {"A": ["-3", "4"], "B": ["10", "9"], "C": ["5", "-3"], "ortho": ["435/131", "179/131"], "orthocenter_approx": [3.3206106870229006, 1.366412213740458], "source_dataset": "advanced_geometry", "source_index": 654, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, -9), B=(-2, -8), and C=(0, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.291, -7.618)", "metadata": {"A": ["7", "-9"], "B": ["-2", "-8"], "C": ["0", "4"], "ortho": ["-71/55", "-419/55"], "orthocenter_approx": [-1.290909090909091, -7.618181818181818], "source_dataset": "advanced_geometry", "source_index": 655, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 3), B=(4, -6), and C=(7, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.183", "metadata": {"A": ["5", "3"], "B": ["4", "-6"], "C": ["7", "-2"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(29)/2 + sqrt(82)/2)*sqrt(-sqrt(29)/2 + 5/2 + sqrt(82)/2)*sqrt(-sqrt(82)/2 + 5/2 + sqrt(29)/2)/sqrt(5/2 + sqrt(29)/2 + sqrt(82)/2)", "incircle_radius_approx": 1.1830941030345576, "source_dataset": "advanced_geometry", "source_index": 656, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 10), B=(-9, 7), and C=(-8, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.219", "metadata": {"A": ["-3", "10"], "B": ["-9", "7"], "C": ["-8", "-6"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + 3*sqrt(5)/2 + sqrt(281)/2)*sqrt(-sqrt(281)/2 + 3*sqrt(5)/2 + sqrt(170)/2)*sqrt(-3*sqrt(5)/2 + sqrt(170)/2 + sqrt(281)/2)/sqrt(3*sqrt(5)/2 + sqrt(170)/2 + sqrt(281)/2)", "incircle_radius_approx": 2.2185907113862346, "source_dataset": "advanced_geometry", "source_index": 657, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 4), B=(-4, 6), and C=(9, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.846, 0.000)", "metadata": {"A": ["-4", "4"], "B": ["-4", "6"], "C": ["9", "0"], "ortho": ["-76/13", "0"], "orthocenter_approx": [-5.846153846153846, 0.0], "source_dataset": "advanced_geometry", "source_index": 658, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, 5), B=(-7, -7), and C=(7, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "20.41°", "metadata": {"A": ["-2", "5"], "B": ["-7", "-7"], "C": ["7", "8"], "angle_ABC_degrees": 20.4052010410776, "source_dataset": "advanced_geometry", "source_index": 659, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -10), B=(-2, -4), and C=(-1, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(26.000, -14.000)", "metadata": {"A": ["-6", "-10"], "B": ["-2", "-4"], "C": ["-1", "4"], "ortho": ["26", "-14"], "orthocenter_approx": [26.0, -14.0], "source_dataset": "advanced_geometry", "source_index": 660, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, -6), B=(5, 10), and C=(-3, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.710, 5.097)", "metadata": {"A": ["0", "-6"], "B": ["5", "10"], "C": ["-3", "3"], "ortho": ["-301/31", "158/31"], "orthocenter_approx": [-9.709677419354838, 5.096774193548387], "source_dataset": "advanced_geometry", "source_index": 661, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, 8), B=(-1, 2), and C=(3, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "131.82°", "metadata": {"A": ["1", "8"], "B": ["-1", "2"], "C": ["3", "-5"], "angle_ABC_degrees": 131.82016988013578, "source_dataset": "advanced_geometry", "source_index": 662, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -1), B=(-1, 3), and C=(5, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.871", "metadata": {"A": ["-10", "-1"], "B": ["-1", "3"], "C": ["5", "-1"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(13) + sqrt(97)/2)*sqrt(-sqrt(13) + sqrt(97)/2 + 15/2)*sqrt(-sqrt(97)/2 + sqrt(13) + 15/2)/sqrt(sqrt(13) + sqrt(97)/2 + 15/2)", "incircle_radius_approx": 1.8714932688587027, "source_dataset": "advanced_geometry", "source_index": 663, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, 6), B=(8, 7), and C=(5, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "72.19°", "metadata": {"A": ["-2", "6"], "B": ["8", "7"], "C": ["5", "-7"], "angle_ABC_degrees": 72.19464978548825, "source_dataset": "advanced_geometry", "source_index": 664, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, 10), B=(0, 6), and C=(7, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.174, 3.609)", "metadata": {"A": ["9", "10"], "B": ["0", "6"], "C": ["7", "4"], "ortho": ["165/23", "83/23"], "orthocenter_approx": [7.173913043478261, 3.608695652173913], "source_dataset": "advanced_geometry", "source_index": 665, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, 1), B=(-1, -6), and C=(1, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.728, 1.570)", "metadata": {"A": ["10", "1"], "B": ["-1", "-6"], "C": ["1", "9"], "ortho": ["865/151", "237/151"], "orthocenter_approx": [5.728476821192053, 1.5695364238410596], "source_dataset": "advanced_geometry", "source_index": 666, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, -9), B=(-3, -7), and C=(-5, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "116.57°", "metadata": {"A": ["-2", "-9"], "B": ["-3", "-7"], "C": ["-5", "-7"], "angle_ABC_degrees": 116.56505117707799, "source_dataset": "advanced_geometry", "source_index": 667, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, 4), B=(-4, 4), and C=(-1, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.387", "metadata": {"A": ["7", "4"], "B": ["-4", "4"], "C": ["-1", "7"], "incircle_radius_exact": "sqrt(-11/2 + 3*sqrt(2)/2 + sqrt(73)/2)*sqrt(-3*sqrt(2)/2 + sqrt(73)/2 + 11/2)*sqrt(-sqrt(73)/2 + 3*sqrt(2)/2 + 11/2)/sqrt(3*sqrt(2)/2 + sqrt(73)/2 + 11/2)", "incircle_radius_approx": 1.3873331353538598, "source_dataset": "advanced_geometry", "source_index": 668, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(5, 2), B=(3, 1), and C=(-3, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "135.00°", "metadata": {"A": ["5", "2"], "B": ["3", "1"], "C": ["-3", "3"], "angle_ABC_degrees": 135.0, "source_dataset": "advanced_geometry", "source_index": 669, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, 5), B=(5, -8), and C=(3, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.454", "metadata": {"A": ["9", "5"], "B": ["5", "-8"], "C": ["3", "-4"], "incircle_radius_exact": "sqrt(-3*sqrt(13)/2 + sqrt(5) + sqrt(185)/2)*sqrt(-sqrt(185)/2 + sqrt(5) + 3*sqrt(13)/2)*sqrt(-sqrt(5) + 3*sqrt(13)/2 + sqrt(185)/2)/sqrt(sqrt(5) + 3*sqrt(13)/2 + sqrt(185)/2)", "incircle_radius_approx": 1.4537771407463973, "source_dataset": "advanced_geometry", "source_index": 670, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, -1), B=(1, 5), and C=(2, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.219", "metadata": {"A": ["10", "-1"], "B": ["1", "5"], "C": ["2", "-8"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + 3*sqrt(13)/2 + sqrt(170)/2)*sqrt(-3*sqrt(13)/2 + sqrt(113)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + sqrt(113)/2 + 3*sqrt(13)/2)/sqrt(sqrt(113)/2 + 3*sqrt(13)/2 + sqrt(170)/2)", "incircle_radius_approx": 3.2187716956251555, "source_dataset": "advanced_geometry", "source_index": 671, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, 0), B=(-7, -1), and C=(3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-52.333, -63.333)", "metadata": {"A": ["-8", "0"], "B": ["-7", "-1"], "C": ["3", "-8"], "ortho": ["-157/3", "-190/3"], "orthocenter_approx": [-52.333333333333336, -63.333333333333336], "source_dataset": "advanced_geometry", "source_index": 672, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, -6), B=(-3, -8), and C=(-9, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.76°", "metadata": {"A": ["-8", "-6"], "B": ["-3", "-8"], "C": ["-9", "9"], "angle_ABC_degrees": 48.758555685472, "source_dataset": "advanced_geometry", "source_index": 673, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, -3), B=(-6, -7), and C=(5, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.833, -2.083)", "metadata": {"A": ["4", "-3"], "B": ["-6", "-7"], "C": ["5", "-5"], "ortho": ["23/6", "-25/12"], "orthocenter_approx": [3.8333333333333335, -2.0833333333333335], "source_dataset": "advanced_geometry", "source_index": 674, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-6, 9), B=(-3, 5), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(19.667, 9.000)", "metadata": {"A": ["-6", "9"], "B": ["-3", "5"], "C": ["-3", "-8"], "ortho": ["59/3", "9"], "orthocenter_approx": [19.666666666666668, 9.0], "source_dataset": "advanced_geometry", "source_index": 675, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, -3), B=(5, 3), and C=(-6, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.993", "metadata": {"A": ["10", "-3"], "B": ["5", "3"], "C": ["-6", "2"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + sqrt(61)/2 + sqrt(281)/2)*sqrt(-sqrt(281)/2 + sqrt(61)/2 + sqrt(122)/2)*sqrt(-sqrt(61)/2 + sqrt(122)/2 + sqrt(281)/2)/sqrt(sqrt(61)/2 + sqrt(122)/2 + sqrt(281)/2)", "incircle_radius_approx": 1.993336903204529, "source_dataset": "advanced_geometry", "source_index": 676, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -5), B=(6, -2), and C=(9, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.511", "metadata": {"A": ["-7", "-5"], "B": ["6", "-2"], "C": ["9", "3"], "incircle_radius_exact": "sqrt(-sqrt(178)/2 + sqrt(34)/2 + 4*sqrt(5))*sqrt(-sqrt(34)/2 + sqrt(178)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(34)/2 + sqrt(178)/2)/sqrt(sqrt(34)/2 + sqrt(178)/2 + 4*sqrt(5))", "incircle_radius_approx": 1.5110158541713186, "source_dataset": "advanced_geometry", "source_index": 677, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, -10), B=(-4, 4), and C=(4, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "41.82°", "metadata": {"A": ["-6", "-10"], "B": ["-4", "4"], "C": ["4", "-8"], "angle_ABC_degrees": 41.820169880135765, "source_dataset": "advanced_geometry", "source_index": 678, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, -5), B=(1, 5), and C=(10, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.535, 1.907)", "metadata": {"A": ["3", "-5"], "B": ["1", "5"], "C": ["10", "3"], "ortho": ["195/43", "82/43"], "orthocenter_approx": [4.534883720930233, 1.9069767441860466], "source_dataset": "advanced_geometry", "source_index": 679, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -5), B=(-1, 7), and C=(3, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "46.58°", "metadata": {"A": ["-6", "-5"], "B": ["-1", "7"], "C": ["3", "-2"], "angle_ABC_degrees": 46.58235392261861, "source_dataset": "advanced_geometry", "source_index": 680, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, -4), B=(-2, -4), and C=(-2, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "90.00°", "metadata": {"A": ["-5", "-4"], "B": ["-2", "-4"], "C": ["-2", "-2"], "angle_ABC_degrees": 90.0, "source_dataset": "advanced_geometry", "source_index": 681, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -9), B=(-7, -1), and C=(-4, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "106.26°", "metadata": {"A": ["-1", "-9"], "B": ["-7", "-1"], "C": ["-4", "3"], "angle_ABC_degrees": 106.26020470831196, "source_dataset": "advanced_geometry", "source_index": 682, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, -9), B=(-10, 7), and C=(7, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(9.778, 2.605)", "metadata": {"A": ["5", "-9"], "B": ["-10", "7"], "C": ["7", "0"], "ortho": ["1633/167", "435/167"], "orthocenter_approx": [9.778443113772456, 2.6047904191616764], "source_dataset": "advanced_geometry", "source_index": 683, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 3), B=(-8, 2), and C=(10, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.170", "metadata": {"A": ["1", "3"], "B": ["-8", "2"], "C": ["10", "-6"], "incircle_radius_exact": "sqrt(-9*sqrt(2)/2 + sqrt(82)/2 + sqrt(97))*sqrt(-sqrt(82)/2 + 9*sqrt(2)/2 + sqrt(97))*sqrt(-sqrt(97) + sqrt(82)/2 + 9*sqrt(2)/2)/sqrt(sqrt(82)/2 + 9*sqrt(2)/2 + sqrt(97))", "incircle_radius_approx": 2.1696668480725405, "source_dataset": "advanced_geometry", "source_index": 684, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, 6), B=(-9, 5), and C=(-5, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.96°", "metadata": {"A": ["-5", "6"], "B": ["-9", "5"], "C": ["-5", "9"], "angle_ABC_degrees": 30.96375653207352, "source_dataset": "advanced_geometry", "source_index": 685, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 9), B=(7, 10), and C=(-7, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-18.609, 24.826)", "metadata": {"A": ["4", "9"], "B": ["7", "10"], "C": ["-7", "-10"], "ortho": ["-428/23", "571/23"], "orthocenter_approx": [-18.608695652173914, 24.82608695652174], "source_dataset": "advanced_geometry", "source_index": 686, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(0, -1), B=(8, -7), and C=(8, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "53.13°", "metadata": {"A": ["0", "-1"], "B": ["8", "-7"], "C": ["8", "-3"], "angle_ABC_degrees": 53.13010235415598, "source_dataset": "advanced_geometry", "source_index": 687, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 9), B=(-1, 8), and C=(7, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.705", "metadata": {"A": ["-7", "9"], "B": ["-1", "8"], "C": ["7", "10"], "incircle_radius_exact": "sqrt(-sqrt(17) + sqrt(37)/2 + sqrt(197)/2)*sqrt(-sqrt(197)/2 + sqrt(37)/2 + sqrt(17))*sqrt(-sqrt(37)/2 + sqrt(17) + sqrt(197)/2)/sqrt(sqrt(37)/2 + sqrt(17) + sqrt(197)/2)", "incircle_radius_approx": 0.7051031899638677, "source_dataset": "advanced_geometry", "source_index": 688, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, 7), B=(-7, -10), and C=(2, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "36.10°", "metadata": {"A": ["-8", "7"], "B": ["-7", "-10"], "C": ["2", "4"], "angle_ABC_degrees": 36.1016869355374, "source_dataset": "advanced_geometry", "source_index": 689, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, 2), B=(10, 10), and C=(10, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "64.80°", "metadata": {"A": ["-7", "2"], "B": ["10", "10"], "C": ["10", "-3"], "angle_ABC_degrees": 64.79887635452492, "source_dataset": "advanced_geometry", "source_index": 690, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, 4), B=(-6, 7), and C=(-6, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.320", "metadata": {"A": ["10", "4"], "B": ["-6", "7"], "C": ["-6", "10"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(265)/2 + sqrt(73))*sqrt(-sqrt(265)/2 + 3/2 + sqrt(73))*sqrt(-sqrt(73) + 3/2 + sqrt(265)/2)/sqrt(3/2 + sqrt(265)/2 + sqrt(73))", "incircle_radius_approx": 1.3198841506199044, "source_dataset": "advanced_geometry", "source_index": 691, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, -2), B=(1, 7), and C=(10, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "43.21°", "metadata": {"A": ["0", "-2"], "B": ["1", "7"], "C": ["10", "-5"], "angle_ABC_degrees": 43.210089391753925, "source_dataset": "advanced_geometry", "source_index": 692, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, -10), B=(7, 5), and C=(-8, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.473", "metadata": {"A": ["-6", "-10"], "B": ["7", "5"], "C": ["-8", "-4"], "incircle_radius_exact": "sqrt(-sqrt(10) + 3*sqrt(34)/2 + sqrt(394)/2)*sqrt(-3*sqrt(34)/2 + sqrt(10) + sqrt(394)/2)*sqrt(-sqrt(394)/2 + sqrt(10) + 3*sqrt(34)/2)/sqrt(sqrt(10) + 3*sqrt(34)/2 + sqrt(394)/2)", "incircle_radius_approx": 2.4732723846770193, "source_dataset": "advanced_geometry", "source_index": 693, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, -8), B=(-9, 5), and C=(-7, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.862", "metadata": {"A": ["-8", "-8"], "B": ["-9", "5"], "C": ["-7", "2"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(13)/2 + sqrt(170)/2)*sqrt(-sqrt(13)/2 + sqrt(101)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + sqrt(13)/2 + sqrt(101)/2)/sqrt(sqrt(13)/2 + sqrt(101)/2 + sqrt(170)/2)", "incircle_radius_approx": 0.8616222748559963, "source_dataset": "advanced_geometry", "source_index": 694, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, -4), B=(-10, -4), and C=(-4, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.000, 14.000)", "metadata": {"A": ["8", "-4"], "B": ["-10", "-4"], "C": ["-4", "0"], "ortho": ["-4", "14"], "orthocenter_approx": [-4.0, 14.0], "source_dataset": "advanced_geometry", "source_index": 695, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, -2), B=(-5, -6), and C=(10, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.667, -12.000)", "metadata": {"A": ["-8", "-2"], "B": ["-5", "-6"], "C": ["10", "-1"], "ortho": ["-14/3", "-12"], "orthocenter_approx": [-4.666666666666667, -12.0], "source_dataset": "advanced_geometry", "source_index": 696, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, 5), B=(-5, -6), and C=(-9, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(13.688, -20.375)", "metadata": {"A": ["1", "5"], "B": ["-5", "-6"], "C": ["-9", "-8"], "ortho": ["219/16", "-163/8"], "orthocenter_approx": [13.6875, -20.375], "source_dataset": "advanced_geometry", "source_index": 697, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, -1), B=(-3, -5), and C=(-8, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "55.49°", "metadata": {"A": ["-9", "-1"], "B": ["-3", "-5"], "C": ["-8", "-7"], "angle_ABC_degrees": 55.49147701233159, "source_dataset": "advanced_geometry", "source_index": 698, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, -10), B=(3, 4), and C=(1, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.062, -8.656)", "metadata": {"A": ["6", "-10"], "B": ["3", "4"], "C": ["1", "-8"], "ortho": ["-33/16", "-277/32"], "orthocenter_approx": [-2.0625, -8.65625], "source_dataset": "advanced_geometry", "source_index": 699, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, -6), B=(8, -7), and C=(9, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.426", "metadata": {"A": ["-1", "-6"], "B": ["8", "-7"], "C": ["9", "10"], "incircle_radius_exact": "sqrt(-sqrt(290)/2 + sqrt(82)/2 + sqrt(89))*sqrt(-sqrt(82)/2 + sqrt(290)/2 + sqrt(89))*sqrt(-sqrt(89) + sqrt(82)/2 + sqrt(290)/2)/sqrt(sqrt(82)/2 + sqrt(290)/2 + sqrt(89))", "incircle_radius_approx": 3.425820569538305, "source_dataset": "advanced_geometry", "source_index": 700, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 8), B=(-9, -10), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "16.27°", "metadata": {"A": ["-1", "8"], "B": ["-9", "-10"], "C": ["2", "3"], "angle_ABC_degrees": 16.27386933469564, "source_dataset": "advanced_geometry", "source_index": 701, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, 0), B=(6, -7), and C=(-10, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.858", "metadata": {"A": ["7", "0"], "B": ["6", "-7"], "C": ["-10", "1"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(290)/2 + 4*sqrt(5))*sqrt(-sqrt(290)/2 + 5*sqrt(2)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + 5*sqrt(2)/2 + sqrt(290)/2)/sqrt(5*sqrt(2)/2 + sqrt(290)/2 + 4*sqrt(5))", "incircle_radius_approx": 2.857891488773211, "source_dataset": "advanced_geometry", "source_index": 702, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 1), B=(-5, -10), and C=(3, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(9.638, -0.241)", "metadata": {"A": ["7", "1"], "B": ["-5", "-10"], "C": ["3", "7"], "ortho": ["559/58", "-7/29"], "orthocenter_approx": [9.637931034482758, -0.2413793103448276], "source_dataset": "advanced_geometry", "source_index": 703, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, -5), B=(2, 4), and C=(8, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.267", "metadata": {"A": ["6", "-5"], "B": ["2", "4"], "C": ["8", "6"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(97)/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + sqrt(10) + sqrt(97)/2)*sqrt(-sqrt(97)/2 + sqrt(10) + 5*sqrt(5)/2)/sqrt(sqrt(10) + sqrt(97)/2 + 5*sqrt(5)/2)", "incircle_radius_approx": 2.2665993941733897, "source_dataset": "advanced_geometry", "source_index": 704, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, -7), B=(4, -2), and C=(-1, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.671", "metadata": {"A": ["10", "-7"], "B": ["4", "-2"], "C": ["-1", "6"], "incircle_radius_exact": "sqrt(-sqrt(290)/2 + sqrt(61)/2 + sqrt(89)/2)*sqrt(-sqrt(61)/2 + sqrt(89)/2 + sqrt(290)/2)*sqrt(-sqrt(89)/2 + sqrt(61)/2 + sqrt(290)/2)/sqrt(sqrt(61)/2 + sqrt(89)/2 + sqrt(290)/2)", "incircle_radius_approx": 0.6710701086292656, "source_dataset": "advanced_geometry", "source_index": 705, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-5, 0), B=(-1, 2), and C=(-9, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.000, -16.000)", "metadata": {"A": ["-5", "0"], "B": ["-1", "2"], "C": ["-9", "0"], "ortho": ["-1", "-16"], "orthocenter_approx": [-1.0, -16.0], "source_dataset": "advanced_geometry", "source_index": 706, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, 10), B=(-5, 3), and C=(0, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.166", "metadata": {"A": ["4", "10"], "B": ["-5", "3"], "C": ["0", "10"], "incircle_radius_exact": "sqrt(-2 + sqrt(74)/2 + sqrt(130)/2)*sqrt(-sqrt(130)/2 + 2 + sqrt(74)/2)*sqrt(-sqrt(74)/2 + 2 + sqrt(130)/2)/sqrt(2 + sqrt(74)/2 + sqrt(130)/2)", "incircle_radius_approx": 1.1664683904652082, "source_dataset": "advanced_geometry", "source_index": 707, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -6), B=(-3, 4), and C=(8, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.362", "metadata": {"A": ["3", "-6"], "B": ["-3", "4"], "C": ["8", "-7"], "incircle_radius_exact": "sqrt(-11*sqrt(2)/2 + sqrt(26)/2 + sqrt(34))*sqrt(-sqrt(26)/2 + sqrt(34) + 11*sqrt(2)/2)*sqrt(-sqrt(34) + sqrt(26)/2 + 11*sqrt(2)/2)/sqrt(sqrt(26)/2 + sqrt(34) + 11*sqrt(2)/2)", "incircle_radius_approx": 1.361501036773726, "source_dataset": "advanced_geometry", "source_index": 708, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -7), B=(0, 3), and C=(7, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "35.84°", "metadata": {"A": ["6", "-7"], "B": ["0", "3"], "C": ["7", "0"], "angle_ABC_degrees": 35.83765295427829, "source_dataset": "advanced_geometry", "source_index": 709, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, 0), B=(2, -8), and C=(10, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "107.10°", "metadata": {"A": ["-9", "0"], "B": ["2", "-8"], "C": ["10", "-2"], "angle_ABC_degrees": 107.10272896905236, "source_dataset": "advanced_geometry", "source_index": 710, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 0), B=(-5, -9), and C=(7, -9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.771", "metadata": {"A": ["-8", "0"], "B": ["-5", "-9"], "C": ["7", "-9"], "incircle_radius_exact": "sqrt(-6 + 3*sqrt(10)/2 + 3*sqrt(34)/2)*sqrt(-3*sqrt(10)/2 + 6 + 3*sqrt(34)/2)*sqrt(-3*sqrt(34)/2 + 3*sqrt(10)/2 + 6)/sqrt(3*sqrt(10)/2 + 6 + 3*sqrt(34)/2)", "incircle_radius_approx": 2.7706737457054107, "source_dataset": "advanced_geometry", "source_index": 711, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, -6), B=(10, -5), and C=(-7, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(22.429, -154.143)", "metadata": {"A": ["5", "-6"], "B": ["10", "-5"], "C": ["-7", "-7"], "ortho": ["157/7", "-1079/7"], "orthocenter_approx": [22.428571428571427, -154.14285714285714], "source_dataset": "advanced_geometry", "source_index": 712, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, 0), B=(-1, -8), and C=(-1, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "20.56°", "metadata": {"A": ["2", "0"], "B": ["-1", "-8"], "C": ["-1", "9"], "angle_ABC_degrees": 20.556045219583464, "source_dataset": "advanced_geometry", "source_index": 713, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, 9), B=(9, 1), and C=(-7, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.476", "metadata": {"A": ["-2", "9"], "B": ["9", "1"], "C": ["-7", "-7"], "incircle_radius_exact": "sqrt(-sqrt(185)/2 + sqrt(281)/2 + 4*sqrt(5))*sqrt(-sqrt(281)/2 + sqrt(185)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(185)/2 + sqrt(281)/2)/sqrt(sqrt(185)/2 + sqrt(281)/2 + 4*sqrt(5))", "incircle_radius_approx": 4.476399216291741, "source_dataset": "advanced_geometry", "source_index": 714, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -5), B=(4, -4), and C=(4, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.806", "metadata": {"A": ["7", "-5"], "B": ["4", "-4"], "C": ["4", "-1"], "incircle_radius_exact": "sqrt(-1 + sqrt(10)/2)*sqrt(1 + sqrt(10)/2)*sqrt(4 - sqrt(10)/2)/sqrt(sqrt(10)/2 + 4)", "incircle_radius_approx": 0.8062870566386034, "source_dataset": "advanced_geometry", "source_index": 715, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 6), B=(-2, 6), and C=(5, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.230", "metadata": {"A": ["-8", "6"], "B": ["-2", "6"], "C": ["5", "7"], "incircle_radius_exact": "sqrt(-3 + 5*sqrt(2)/2 + sqrt(170)/2)*sqrt(-sqrt(170)/2 + 3 + 5*sqrt(2)/2)*sqrt(-5*sqrt(2)/2 + 3 + sqrt(170)/2)/sqrt(3 + 5*sqrt(2)/2 + sqrt(170)/2)", "incircle_radius_approx": 0.22980165424261154, "source_dataset": "advanced_geometry", "source_index": 716, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, -8), B=(0, -4), and C=(4, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.000, -28.000)", "metadata": {"A": ["7", "-8"], "B": ["0", "-4"], "C": ["4", "-7"], "ortho": ["-8", "-28"], "orthocenter_approx": [-8.0, -28.0], "source_dataset": "advanced_geometry", "source_index": 717, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, 0), B=(-7, -4), and C=(-10, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, 2.000)", "metadata": {"A": ["-7", "0"], "B": ["-7", "-4"], "C": ["-10", "2"], "ortho": ["-3", "2"], "orthocenter_approx": [-3.0, 2.0], "source_dataset": "advanced_geometry", "source_index": 718, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, -3), B=(9, -2), and C=(8, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(15.000, 0.000)", "metadata": {"A": ["9", "-3"], "B": ["9", "-2"], "C": ["8", "0"], "ortho": ["15", "0"], "orthocenter_approx": [15.0, 0.0], "source_dataset": "advanced_geometry", "source_index": 719, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, -5), B=(2, -3), and C=(-8, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "80.13°", "metadata": {"A": ["1", "-5"], "B": ["2", "-3"], "C": ["-8", "0"], "angle_ABC_degrees": 80.13419305691563, "source_dataset": "advanced_geometry", "source_index": 720, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, 5), B=(-7, -9), and C=(6, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "93.35°", "metadata": {"A": ["-10", "5"], "B": ["-7", "-9"], "C": ["6", "-7"], "angle_ABC_degrees": 93.34859481445689, "source_dataset": "advanced_geometry", "source_index": 721, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, -8), B=(3, 10), and C=(6, 2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "44.52°", "metadata": {"A": ["-5", "-8"], "B": ["3", "10"], "C": ["6", "2"], "angle_ABC_degrees": 44.51853419416165, "source_dataset": "advanced_geometry", "source_index": 722, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, -3), B=(3, -2), and C=(-5, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.143, 7.286)", "metadata": {"A": ["5", "-3"], "B": ["3", "-2"], "C": ["-5", "-5"], "ortho": ["8/7", "51/7"], "orthocenter_approx": [1.1428571428571428, 7.285714285714286], "source_dataset": "advanced_geometry", "source_index": 723, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, 1), B=(-2, 0), and C=(-6, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.547", "metadata": {"A": ["-3", "1"], "B": ["-2", "0"], "C": ["-6", "-1"], "incircle_radius_exact": "sqrt(-sqrt(13)/2 + sqrt(2)/2 + sqrt(17)/2)*sqrt(-sqrt(17)/2 + sqrt(2)/2 + sqrt(13)/2)*sqrt(-sqrt(2)/2 + sqrt(13)/2 + sqrt(17)/2)/sqrt(sqrt(2)/2 + sqrt(13)/2 + sqrt(17)/2)", "incircle_radius_approx": 0.5468742032369, "source_dataset": "advanced_geometry", "source_index": 724, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -7), B=(10, 5), and C=(-4, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.13°", "metadata": {"A": ["-7", "-7"], "B": ["10", "5"], "C": ["-4", "4"], "angle_ABC_degrees": 31.131976188217838, "source_dataset": "advanced_geometry", "source_index": 725, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-2, 4), B=(-1, -3), and C=(5, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.923, -7.846)", "metadata": {"A": ["-2", "4"], "B": ["-1", "-3"], "C": ["5", "-6"], "ortho": ["-103/13", "-102/13"], "orthocenter_approx": [-7.923076923076923, -7.846153846153846], "source_dataset": "advanced_geometry", "source_index": 726, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 9), B=(10, -7), and C=(7, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(29.222, 21.833)", "metadata": {"A": ["-5", "9"], "B": ["10", "-7"], "C": ["7", "1"], "ortho": ["263/9", "131/6"], "orthocenter_approx": [29.22222222222222, 21.833333333333332], "source_dataset": "advanced_geometry", "source_index": 727, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 4), B=(-6, -9), and C=(0, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.366", "metadata": {"A": ["-5", "4"], "B": ["-6", "-9"], "C": ["0", "-9"], "incircle_radius_exact": "sqrt(-3 + sqrt(170)/2 + sqrt(194)/2)*sqrt(-sqrt(170)/2 + 3 + sqrt(194)/2)*sqrt(-sqrt(194)/2 + 3 + sqrt(170)/2)/sqrt(3 + sqrt(170)/2 + sqrt(194)/2)", "incircle_radius_approx": 2.3660172159530934, "source_dataset": "advanced_geometry", "source_index": 728, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -5), B=(4, 2), and C=(-8, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.859", "metadata": {"A": ["-6", "-5"], "B": ["4", "2"], "C": ["-8", "-1"], "incircle_radius_exact": "sqrt(-sqrt(149)/2 + sqrt(5) + 3*sqrt(17)/2)*sqrt(-3*sqrt(17)/2 + sqrt(5) + sqrt(149)/2)*sqrt(-sqrt(5) + sqrt(149)/2 + 3*sqrt(17)/2)/sqrt(sqrt(5) + sqrt(149)/2 + 3*sqrt(17)/2)", "incircle_radius_approx": 1.8589914725973964, "source_dataset": "advanced_geometry", "source_index": 729, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 0), B=(-9, -4), and C=(-6, 0), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.000, 12.000)", "metadata": {"A": ["7", "0"], "B": ["-9", "-4"], "C": ["-6", "0"], "ortho": ["-9", "12"], "orthocenter_approx": [-9.0, 12.0], "source_dataset": "advanced_geometry", "source_index": 730, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, 1), B=(0, 7), and C=(5, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(8.600, 7.000)", "metadata": {"A": ["5", "1"], "B": ["0", "7"], "C": ["5", "4"], "ortho": ["43/5", "7"], "orthocenter_approx": [8.6, 7.0], "source_dataset": "advanced_geometry", "source_index": 731, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -7), B=(8, 5), and C=(4, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.836", "metadata": {"A": ["-10", "-7"], "B": ["8", "5"], "C": ["4", "-5"], "incircle_radius_exact": "sqrt(-3*sqrt(13) + sqrt(29) + 5*sqrt(2))*sqrt(-5*sqrt(2) + sqrt(29) + 3*sqrt(13))*sqrt(-sqrt(29) + 5*sqrt(2) + 3*sqrt(13))/sqrt(sqrt(29) + 5*sqrt(2) + 3*sqrt(13))", "incircle_radius_approx": 2.83591810387869, "source_dataset": "advanced_geometry", "source_index": 732, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, -5), B=(10, 3), and C=(3, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.061", "metadata": {"A": ["-10", "-5"], "B": ["10", "3"], "C": ["3", "-10"], "incircle_radius_exact": "sqrt(-sqrt(194)/2 + sqrt(218)/2 + 2*sqrt(29))*sqrt(-sqrt(218)/2 + sqrt(194)/2 + 2*sqrt(29))*sqrt(-2*sqrt(29) + sqrt(194)/2 + sqrt(218)/2)/sqrt(sqrt(194)/2 + sqrt(218)/2 + 2*sqrt(29))", "incircle_radius_approx": 4.061005009202988, "source_dataset": "advanced_geometry", "source_index": 733, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 3), B=(-7, 5), and C=(-10, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.050", "metadata": {"A": ["9", "3"], "B": ["-7", "5"], "C": ["-10", "8"], "incircle_radius_exact": "sqrt(-3*sqrt(2)/2 + sqrt(65) + sqrt(386)/2)*sqrt(-sqrt(386)/2 + 3*sqrt(2)/2 + sqrt(65))*sqrt(-sqrt(65) + 3*sqrt(2)/2 + sqrt(386)/2)/sqrt(3*sqrt(2)/2 + sqrt(65) + sqrt(386)/2)", "incircle_radius_approx": 1.049631608482429, "source_dataset": "advanced_geometry", "source_index": 734, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, 4), B=(1, 8), and C=(-2, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.556, 4.167)", "metadata": {"A": ["4", "4"], "B": ["1", "8"], "C": ["-2", "0"], "ortho": ["32/9", "25/6"], "orthocenter_approx": [3.5555555555555554, 4.166666666666667], "source_dataset": "advanced_geometry", "source_index": 735, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(1, -6), B=(1, -4), and C=(3, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-111.000, 10.000)", "metadata": {"A": ["1", "-6"], "B": ["1", "-4"], "C": ["3", "10"], "ortho": ["-111", "10"], "orthocenter_approx": [-111.0, 10.0], "source_dataset": "advanced_geometry", "source_index": 736, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(7, -5), B=(2, 6), and C=(-4, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "58.13°", "metadata": {"A": ["7", "-5"], "B": ["2", "6"], "C": ["-4", "-3"], "angle_ABC_degrees": 58.134022306396325, "source_dataset": "advanced_geometry", "source_index": 737, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, -5), B=(4, -4), and C=(3, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.828, -4.310)", "metadata": {"A": ["0", "-5"], "B": ["4", "-4"], "C": ["3", "3"], "ortho": ["140/29", "-125/29"], "orthocenter_approx": [4.827586206896552, -4.310344827586206], "source_dataset": "advanced_geometry", "source_index": 738, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-3, 2), B=(-6, 0), and C=(5, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.304, 2.957)", "metadata": {"A": ["-3", "2"], "B": ["-6", "0"], "C": ["5", "-8"], "ortho": ["-53/23", "68/23"], "orthocenter_approx": [-2.3043478260869565, 2.9565217391304346], "source_dataset": "advanced_geometry", "source_index": 739, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -2), B=(0, 6), and C=(1, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "67.75°", "metadata": {"A": ["-7", "-2"], "B": ["0", "6"], "C": ["1", "4"], "angle_ABC_degrees": 67.75097634278764, "source_dataset": "advanced_geometry", "source_index": 740, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, -4), B=(-4, 4), and C=(2, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "13.39°", "metadata": {"A": ["9", "-4"], "B": ["-4", "4"], "C": ["2", "-2"], "angle_ABC_degrees": 13.392497753751098, "source_dataset": "advanced_geometry", "source_index": 741, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, -9), B=(-1, -3), and C=(-1, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.818", "metadata": {"A": ["9", "-9"], "B": ["-1", "-3"], "C": ["-1", "10"], "incircle_radius_exact": "sqrt(-13/2 + sqrt(34) + sqrt(461)/2)*sqrt(-sqrt(34) + 13/2 + sqrt(461)/2)*sqrt(-sqrt(461)/2 + sqrt(34) + 13/2)/sqrt(sqrt(34) + 13/2 + sqrt(461)/2)", "incircle_radius_approx": 2.817950776483511, "source_dataset": "advanced_geometry", "source_index": 742, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -5), B=(2, -10), and C=(-6, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "36.98°", "metadata": {"A": ["-7", "-5"], "B": ["2", "-10"], "C": ["-6", "8"], "angle_ABC_degrees": 36.982906926344675, "source_dataset": "advanced_geometry", "source_index": 743, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, -2), B=(10, -4), and C=(-5, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.034", "metadata": {"A": ["-9", "-2"], "B": ["10", "-4"], "C": ["-5", "8"], "incircle_radius_exact": "sqrt(-sqrt(29) + sqrt(365)/2 + 3*sqrt(41)/2)*sqrt(-sqrt(365)/2 + sqrt(29) + 3*sqrt(41)/2)*sqrt(-3*sqrt(41)/2 + sqrt(29) + sqrt(365)/2)/sqrt(sqrt(29) + sqrt(365)/2 + 3*sqrt(41)/2)", "incircle_radius_approx": 4.033845553191464, "source_dataset": "advanced_geometry", "source_index": 744, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, 10), B=(5, -6), and C=(1, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.795", "metadata": {"A": ["5", "10"], "B": ["5", "-6"], "C": ["1", "9"], "incircle_radius_exact": "sqrt(-8 + sqrt(17)/2 + sqrt(241)/2)*sqrt(-sqrt(17)/2 + sqrt(241)/2 + 8)*sqrt(-sqrt(241)/2 + sqrt(17)/2 + 8)/sqrt(sqrt(17)/2 + sqrt(241)/2 + 8)", "incircle_radius_approx": 1.7953683821629864, "source_dataset": "advanced_geometry", "source_index": 745, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(5, -10), B=(1, -5), and C=(10, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["5", "-10"], "B": ["1", "-5"], "C": ["10", "-6"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 746, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, -9), B=(5, -6), and C=(2, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.511", "metadata": {"A": ["10", "-9"], "B": ["5", "-6"], "C": ["2", "7"], "incircle_radius_exact": "sqrt(-sqrt(178)/2 + sqrt(34)/2 + 4*sqrt(5))*sqrt(-sqrt(34)/2 + sqrt(178)/2 + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(34)/2 + sqrt(178)/2)/sqrt(sqrt(34)/2 + sqrt(178)/2 + 4*sqrt(5))", "incircle_radius_approx": 1.5110158541713186, "source_dataset": "advanced_geometry", "source_index": 747, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-1, 4), B=(-8, 5), and C=(7, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.44°", "metadata": {"A": ["-1", "4"], "B": ["-8", "5"], "C": ["7", "8"], "angle_ABC_degrees": 19.44003482817619, "source_dataset": "advanced_geometry", "source_index": 748, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-4, 9), B=(-10, -4), and C=(-5, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "76.53°", "metadata": {"A": ["-4", "9"], "B": ["-10", "-4"], "C": ["-5", "-5"], "angle_ABC_degrees": 76.53479190518829, "source_dataset": "advanced_geometry", "source_index": 749, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, 3), B=(6, -7), and C=(3, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "12.99°", "metadata": {"A": ["6", "3"], "B": ["6", "-7"], "C": ["3", "6"], "angle_ABC_degrees": 12.994616791916505, "source_dataset": "advanced_geometry", "source_index": 750, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, -7), B=(7, -10), and C=(4, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "51.01°", "metadata": {"A": ["-2", "-7"], "B": ["7", "-10"], "C": ["4", "-2"], "angle_ABC_degrees": 51.00900595749453, "source_dataset": "advanced_geometry", "source_index": 751, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 6), B=(-8, -1), and C=(3, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.333", "metadata": {"A": ["-5", "6"], "B": ["-8", "-1"], "C": ["3", "3"], "incircle_radius_exact": "sqrt(-sqrt(137)/2 + sqrt(58)/2 + sqrt(73)/2)*sqrt(-sqrt(58)/2 + sqrt(73)/2 + sqrt(137)/2)*sqrt(-sqrt(73)/2 + sqrt(58)/2 + sqrt(137)/2)/sqrt(sqrt(58)/2 + sqrt(73)/2 + sqrt(137)/2)", "incircle_radius_approx": 2.3327192021374725, "source_dataset": "advanced_geometry", "source_index": 752, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, 5), B=(-5, -1), and C=(-1, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.953", "metadata": {"A": ["4", "5"], "B": ["-5", "-1"], "C": ["-1", "-4"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(106)/2 + 3*sqrt(13)/2)*sqrt(-sqrt(106)/2 + 5/2 + 3*sqrt(13)/2)*sqrt(-3*sqrt(13)/2 + 5/2 + sqrt(106)/2)/sqrt(5/2 + sqrt(106)/2 + 3*sqrt(13)/2)", "incircle_radius_approx": 1.953103760043061, "source_dataset": "advanced_geometry", "source_index": 753, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, 5), B=(7, 9), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.94°", "metadata": {"A": ["4", "5"], "B": ["7", "9"], "C": ["2", "3"], "angle_ABC_degrees": 2.9356734464211725, "source_dataset": "advanced_geometry", "source_index": 754, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -6), B=(-8, 5), and C=(8, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "47.73°", "metadata": {"A": ["2", "-6"], "B": ["-8", "5"], "C": ["8", "5"], "angle_ABC_degrees": 47.72631099390627, "source_dataset": "advanced_geometry", "source_index": 755, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-2, 3), B=(-9, 3), and C=(-3, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.758", "metadata": {"A": ["-2", "3"], "B": ["-9", "3"], "C": ["-3", "-2"], "incircle_radius_exact": "sqrt(-7/2 + sqrt(26)/2 + sqrt(61)/2)*sqrt(-sqrt(26)/2 + 7/2 + sqrt(61)/2)*sqrt(-sqrt(61)/2 + sqrt(26)/2 + 7/2)/sqrt(sqrt(26)/2 + 7/2 + sqrt(61)/2)", "incircle_radius_approx": 1.7579751253983609, "source_dataset": "advanced_geometry", "source_index": 756, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, 2), B=(5, -10), and C=(-5, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.000, -2.167)", "metadata": {"A": ["10", "2"], "B": ["5", "-10"], "C": ["-5", "2"], "ortho": ["5", "-13/6"], "orthocenter_approx": [5.0, -2.1666666666666665], "source_dataset": "advanced_geometry", "source_index": 757, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(5, 8), B=(-9, 5), and C=(4, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "7.70°", "metadata": {"A": ["5", "8"], "B": ["-9", "5"], "C": ["4", "6"], "angle_ABC_degrees": 7.696051722016568, "source_dataset": "advanced_geometry", "source_index": 758, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -8), B=(-6, 0), and C=(6, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(17.556, 17.667)", "metadata": {"A": ["9", "-8"], "B": ["-6", "0"], "C": ["6", "-4"], "ortho": ["158/9", "53/3"], "orthocenter_approx": [17.555555555555557, 17.666666666666668], "source_dataset": "advanced_geometry", "source_index": 759, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, 7), B=(1, 0), and C=(-1, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "113.20°", "metadata": {"A": ["-6", "7"], "B": ["1", "0"], "C": ["-1", "-5"], "angle_ABC_degrees": 113.19859051364818, "source_dataset": "advanced_geometry", "source_index": 760, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 5), B=(6, 3), and C=(-7, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.391, 5.848)", "metadata": {"A": ["-5", "5"], "B": ["6", "3"], "C": ["-7", "-3"], "ortho": ["-124/23", "269/46"], "orthocenter_approx": [-5.391304347826087, 5.8478260869565215], "source_dataset": "advanced_geometry", "source_index": 761, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, -8), B=(0, -10), and C=(-8, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.566", "metadata": {"A": ["4", "-8"], "B": ["0", "-10"], "C": ["-8", "-3"], "incircle_radius_exact": "sqrt(-13/2 + sqrt(5) + sqrt(113)/2)*sqrt(-sqrt(113)/2 + sqrt(5) + 13/2)*sqrt(-sqrt(5) + sqrt(113)/2 + 13/2)/sqrt(sqrt(5) + sqrt(113)/2 + 13/2)", "incircle_radius_approx": 1.565709160688824, "source_dataset": "advanced_geometry", "source_index": 762, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-3, -8), B=(-2, -9), and C=(4, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.641", "metadata": {"A": ["-3", "-8"], "B": ["-2", "-9"], "C": ["4", "4"], "incircle_radius_exact": "sqrt(-sqrt(193)/2 + sqrt(2)/2 + sqrt(205)/2)*sqrt(-sqrt(2)/2 + sqrt(193)/2 + sqrt(205)/2)*sqrt(-sqrt(205)/2 + sqrt(2)/2 + sqrt(193)/2)/sqrt(sqrt(2)/2 + sqrt(193)/2 + sqrt(205)/2)", "incircle_radius_approx": 0.6413614986059508, "source_dataset": "advanced_geometry", "source_index": 763, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 8), B=(-1, 7), and C=(0, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.890", "metadata": {"A": ["-10", "8"], "B": ["-1", "7"], "C": ["0", "9"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(5)/2 + sqrt(82)/2)*sqrt(-sqrt(5)/2 + sqrt(82)/2 + sqrt(101)/2)*sqrt(-sqrt(82)/2 + sqrt(5)/2 + sqrt(101)/2)/sqrt(sqrt(5)/2 + sqrt(82)/2 + sqrt(101)/2)", "incircle_radius_approx": 0.8902913325764288, "source_dataset": "advanced_geometry", "source_index": 764, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-7, -5), B=(-3, -1), and C=(-5, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.534", "metadata": {"A": ["-7", "-5"], "B": ["-3", "-1"], "C": ["-5", "7"], "incircle_radius_exact": "sqrt(-sqrt(17) + 2*sqrt(2) + sqrt(37))*sqrt(-2*sqrt(2) + sqrt(17) + sqrt(37))*sqrt(-sqrt(37) + 2*sqrt(2) + sqrt(17))/sqrt(2*sqrt(2) + sqrt(17) + sqrt(37))", "incircle_radius_approx": 1.5344136042147505, "source_dataset": "advanced_geometry", "source_index": 765, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, 10), B=(-1, -6), and C=(4, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.826", "metadata": {"A": ["4", "10"], "B": ["-1", "-6"], "C": ["4", "-3"], "incircle_radius_exact": "sqrt(-13/2 + sqrt(34)/2 + sqrt(281)/2)*sqrt(-sqrt(281)/2 + sqrt(34)/2 + 13/2)*sqrt(-sqrt(34)/2 + 13/2 + sqrt(281)/2)/sqrt(sqrt(34)/2 + 13/2 + sqrt(281)/2)", "incircle_radius_approx": 1.8261501408504968, "source_dataset": "advanced_geometry", "source_index": 766, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, -5), B=(9, -1), and C=(10, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(16.839, -7.968)", "metadata": {"A": ["2", "-5"], "B": ["9", "-1"], "C": ["10", "4"], "ortho": ["522/31", "-247/31"], "orthocenter_approx": [16.838709677419356, -7.967741935483871], "source_dataset": "advanced_geometry", "source_index": 767, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, 4), B=(-9, -5), and C=(10, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.92°", "metadata": {"A": ["8", "4"], "B": ["-9", "-5"], "C": ["10", "-2"], "angle_ABC_degrees": 18.924644416051233, "source_dataset": "advanced_geometry", "source_index": 768, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-5, 6), B=(6, 3), and C=(-9, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.919, 8.297)", "metadata": {"A": ["-5", "6"], "B": ["6", "3"], "C": ["-9", "-3"], "ortho": ["-219/37", "307/37"], "orthocenter_approx": [-5.918918918918919, 8.297297297297296], "source_dataset": "advanced_geometry", "source_index": 769, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, 9), B=(-1, 1), and C=(10, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.809, 2.447)", "metadata": {"A": ["2", "9"], "B": ["-1", "1"], "C": ["10", "-1"], "ortho": ["38/47", "115/47"], "orthocenter_approx": [0.8085106382978723, 2.4468085106382977], "source_dataset": "advanced_geometry", "source_index": 770, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 9), B=(9, 2), and C=(2, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.240, -8.080)", "metadata": {"A": ["-10", "9"], "B": ["9", "2"], "C": ["2", "-2"], "ortho": ["-6/25", "-202/25"], "orthocenter_approx": [-0.24, -8.08], "source_dataset": "advanced_geometry", "source_index": 771, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -6), B=(-3, 10), and C=(-10, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "24.37°", "metadata": {"A": ["-6", "-6"], "B": ["-3", "10"], "C": ["-10", "0"], "angle_ABC_degrees": 24.372364922403527, "source_dataset": "advanced_geometry", "source_index": 772, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -9), B=(9, -1), and C=(0, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.545, -0.682)", "metadata": {"A": ["-1", "-9"], "B": ["9", "-1"], "C": ["0", "5"], "ortho": ["50/11", "-15/22"], "orthocenter_approx": [4.545454545454545, -0.6818181818181818], "source_dataset": "advanced_geometry", "source_index": 773, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, -7), B=(3, 5), and C=(-8, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(3.944, -3.028)", "metadata": {"A": ["9", "-7"], "B": ["3", "5"], "C": ["-8", "-9"], "ortho": ["71/18", "-109/36"], "orthocenter_approx": [3.9444444444444446, -3.0277777777777777], "source_dataset": "advanced_geometry", "source_index": 774, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(9, 0), B=(1, -4), and C=(-2, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.778, -1.556)", "metadata": {"A": ["9", "0"], "B": ["1", "-4"], "C": ["-2", "8"], "ortho": ["25/9", "-14/9"], "orthocenter_approx": [2.7777777777777777, -1.5555555555555556], "source_dataset": "advanced_geometry", "source_index": 775, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, 0), B=(7, 5), and C=(-8, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.044", "metadata": {"A": ["7", "0"], "B": ["7", "5"], "C": ["-8", "7"], "incircle_radius_exact": "sqrt(-5/2 + sqrt(229)/2 + sqrt(274)/2)*sqrt(-sqrt(229)/2 + 5/2 + sqrt(274)/2)*sqrt(-sqrt(274)/2 + 5/2 + sqrt(229)/2)/sqrt(5/2 + sqrt(229)/2 + sqrt(274)/2)", "incircle_radius_approx": 2.0443938038676883, "source_dataset": "advanced_geometry", "source_index": 776, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, -6), B=(-1, 1), and C=(-6, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.456", "metadata": {"A": ["-5", "-6"], "B": ["-1", "1"], "C": ["-6", "-1"], "incircle_radius_exact": "sqrt(-sqrt(26)/2 + sqrt(29)/2 + sqrt(65)/2)*sqrt(-sqrt(29)/2 + sqrt(26)/2 + sqrt(65)/2)*sqrt(-sqrt(65)/2 + sqrt(26)/2 + sqrt(29)/2)/sqrt(sqrt(26)/2 + sqrt(29)/2 + sqrt(65)/2)", "incircle_radius_approx": 1.455804833051636, "source_dataset": "advanced_geometry", "source_index": 777, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 6), B=(5, -7), and C=(-4, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.000, -3.000)", "metadata": {"A": ["0", "6"], "B": ["5", "-7"], "C": ["-4", "-3"], "ortho": ["-4", "-3"], "orthocenter_approx": [-4.0, -3.0], "source_dataset": "advanced_geometry", "source_index": 778, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, 5), B=(-1, 1), and C=(-3, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "124.02°", "metadata": {"A": ["10", "5"], "B": ["-1", "1"], "C": ["-3", "-7"], "angle_ABC_degrees": 124.01934998982647, "source_dataset": "advanced_geometry", "source_index": 779, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(2, 0), B=(-5, 4), and C=(-1, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(5.667, 14.667)", "metadata": {"A": ["2", "0"], "B": ["-5", "4"], "C": ["-1", "3"], "ortho": ["17/3", "44/3"], "orthocenter_approx": [5.666666666666667, 14.666666666666666], "source_dataset": "advanced_geometry", "source_index": 780, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, -3), B=(-5, -8), and C=(7, 1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.922", "metadata": {"A": ["9", "-3"], "B": ["-5", "-8"], "C": ["7", "1"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(5) + sqrt(221)/2)*sqrt(-sqrt(221)/2 + sqrt(5) + 15/2)*sqrt(-sqrt(5) + sqrt(221)/2 + 15/2)/sqrt(sqrt(5) + sqrt(221)/2 + 15/2)", "incircle_radius_approx": 1.922057386871612, "source_dataset": "advanced_geometry", "source_index": 781, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 7), B=(1, -9), and C=(-6, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "66.37°", "metadata": {"A": ["-6", "7"], "B": ["1", "-9"], "C": ["-6", "-9"], "angle_ABC_degrees": 66.37062226934319, "source_dataset": "advanced_geometry", "source_index": 782, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, -1), B=(8, -7), and C=(10, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.464", "metadata": {"A": ["10", "-1"], "B": ["8", "-7"], "C": ["10", "-4"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(13)/2 + sqrt(10))*sqrt(-sqrt(10) + 3/2 + sqrt(13)/2)*sqrt(-sqrt(13)/2 + 3/2 + sqrt(10))/sqrt(3/2 + sqrt(13)/2 + sqrt(10))", "incircle_radius_approx": 0.46403329744772503, "source_dataset": "advanced_geometry", "source_index": 783, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(10, -2), B=(5, 1), and C=(-6, -4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "124.59°", "metadata": {"A": ["10", "-2"], "B": ["5", "1"], "C": ["-6", "-4"], "angle_ABC_degrees": 124.59228868750995, "source_dataset": "advanced_geometry", "source_index": 784, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-2, 5), B=(9, -9), and C=(6, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.791", "metadata": {"A": ["-2", "5"], "B": ["9", "-9"], "C": ["6", "5"], "incircle_radius_exact": "sqrt(-4 + sqrt(205)/2 + sqrt(317)/2)*sqrt(-sqrt(205)/2 + 4 + sqrt(317)/2)*sqrt(-sqrt(317)/2 + 4 + sqrt(205)/2)/sqrt(4 + sqrt(205)/2 + sqrt(317)/2)", "incircle_radius_approx": 2.7914640603475545, "source_dataset": "advanced_geometry", "source_index": 785, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -8), B=(-3, 8), and C=(-6, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-43.333, 8.000)", "metadata": {"A": ["-6", "-8"], "B": ["-3", "8"], "C": ["-6", "1"], "ortho": ["-130/3", "8"], "orthocenter_approx": [-43.333333333333336, 8.0], "source_dataset": "advanced_geometry", "source_index": 786, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, 5), B=(-4, 8), and C=(-10, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.800, 6.200)", "metadata": {"A": ["-1", "5"], "B": ["-4", "8"], "C": ["-10", "-1"], "ortho": ["-14/5", "31/5"], "orthocenter_approx": [-2.8, 6.2], "source_dataset": "advanced_geometry", "source_index": 787, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, -2), B=(-2, 4), and C=(10, -10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.698, -6.884)", "metadata": {"A": ["-3", "-2"], "B": ["-2", "4"], "C": ["10", "-10"], "ortho": ["-374/43", "-296/43"], "orthocenter_approx": [-8.69767441860465, -6.883720930232558], "source_dataset": "advanced_geometry", "source_index": 788, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, 2), B=(10, -6), and C=(-10, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.000, 6.375)", "metadata": {"A": ["1", "2"], "B": ["10", "-6"], "C": ["-10", "-6"], "ortho": ["1", "51/8"], "orthocenter_approx": [1.0, 6.375], "source_dataset": "advanced_geometry", "source_index": 789, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, 7), B=(0, 6), and C=(8, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "54.08°", "metadata": {"A": ["10", "7"], "B": ["0", "6"], "C": ["8", "-3"], "angle_ABC_degrees": 54.07705380092944, "source_dataset": "advanced_geometry", "source_index": 790, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -1), B=(1, 8), and C=(-5, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "63.43°", "metadata": {"A": ["9", "-1"], "B": ["1", "8"], "C": ["-5", "-7"], "angle_ABC_degrees": 63.43494882292201, "source_dataset": "advanced_geometry", "source_index": 791, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -3), B=(8, -5), and C=(7, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.996", "metadata": {"A": ["-4", "-3"], "B": ["8", "-5"], "C": ["7", "-7"], "incircle_radius_exact": "sqrt(-sqrt(137)/2 + sqrt(5)/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(5)/2 + sqrt(137)/2)*sqrt(-sqrt(5)/2 + sqrt(137)/2 + sqrt(37))/sqrt(sqrt(5)/2 + sqrt(137)/2 + sqrt(37))", "incircle_radius_approx": 0.9959284549122216, "source_dataset": "advanced_geometry", "source_index": 792, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -4), B=(1, -7), and C=(10, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "19.44°", "metadata": {"A": ["10", "-4"], "B": ["1", "-7"], "C": ["10", "0"], "angle_ABC_degrees": 19.44003482817619, "source_dataset": "advanced_geometry", "source_index": 793, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-10, 7), B=(10, -7), and C=(-10, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "5.047", "metadata": {"A": ["-10", "7"], "B": ["10", "-7"], "C": ["-10", "-8"], "incircle_radius_exact": "sqrt(-15/2 + sqrt(401)/2 + sqrt(149))*sqrt(-sqrt(149) + 15/2 + sqrt(401)/2)*sqrt(-sqrt(401)/2 + 15/2 + sqrt(149))/sqrt(15/2 + sqrt(401)/2 + sqrt(149))", "incircle_radius_approx": 5.047268033078294, "source_dataset": "advanced_geometry", "source_index": 794, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, 4), B=(-7, 8), and C=(-2, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "38.66°", "metadata": {"A": ["-2", "4"], "B": ["-7", "8"], "C": ["-2", "8"], "angle_ABC_degrees": 38.65980825409009, "source_dataset": "advanced_geometry", "source_index": 795, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, 0), B=(8, -1), and C=(-8, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "66.37°", "metadata": {"A": ["8", "0"], "B": ["8", "-1"], "C": ["-8", "6"], "angle_ABC_degrees": 66.37062226934319, "source_dataset": "advanced_geometry", "source_index": 796, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, -2), B=(10, -4), and C=(-3, 2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "18.07°", "metadata": {"A": ["-7", "-2"], "B": ["10", "-4"], "C": ["-3", "2"], "angle_ABC_degrees": 18.065303761074986, "source_dataset": "advanced_geometry", "source_index": 797, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, 4), B=(7, 4), and C=(4, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.000, 3.000)", "metadata": {"A": ["3", "4"], "B": ["7", "4"], "C": ["4", "1"], "ortho": ["4", "3"], "orthocenter_approx": [4.0, 3.0], "source_dataset": "advanced_geometry", "source_index": 798, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 2), B=(-4, 8), and C=(-4, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "65.22°", "metadata": {"A": ["9", "2"], "B": ["-4", "8"], "C": ["-4", "6"], "angle_ABC_degrees": 65.22485943116808, "source_dataset": "advanced_geometry", "source_index": 799, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, -5), B=(8, -8), and C=(0, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-31.000, -21.000)", "metadata": {"A": ["5", "-5"], "B": ["8", "-8"], "C": ["0", "10"], "ortho": ["-31", "-21"], "orthocenter_approx": [-31.0, -21.0], "source_dataset": "advanced_geometry", "source_index": 800, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 6), B=(-10, 3), and C=(7, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.62°", "metadata": {"A": ["6", "6"], "B": ["-10", "3"], "C": ["7", "7"], "angle_ABC_degrees": 2.62086463903207, "source_dataset": "advanced_geometry", "source_index": 801, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 7), B=(-3, -7), and C=(9, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.13°", "metadata": {"A": ["6", "7"], "B": ["-3", "-7"], "C": ["9", "9"], "angle_ABC_degrees": 4.134671373736423, "source_dataset": "advanced_geometry", "source_index": 802, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, 10), B=(-8, 7), and C=(-6, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-42.000, 92.000)", "metadata": {"A": ["-1", "10"], "B": ["-8", "7"], "C": ["-6", "8"], "ortho": ["-42", "92"], "orthocenter_approx": [-42.0, 92.0], "source_dataset": "advanced_geometry", "source_index": 803, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-7, 5), B=(-5, -5), and C=(-2, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.000, -6.000)", "metadata": {"A": ["-7", "5"], "B": ["-5", "-5"], "C": ["-2", "-5"], "ortho": ["-7", "-6"], "orthocenter_approx": [-7.0, -6.0], "source_dataset": "advanced_geometry", "source_index": 804, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -5), B=(7, 8), and C=(-1, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.840", "metadata": {"A": ["2", "-5"], "B": ["7", "8"], "C": ["-1", "-7"], "incircle_radius_exact": "sqrt(-17/2 + sqrt(13)/2 + sqrt(194)/2)*sqrt(-sqrt(13)/2 + sqrt(194)/2 + 17/2)*sqrt(-sqrt(194)/2 + sqrt(13)/2 + 17/2)/sqrt(sqrt(13)/2 + sqrt(194)/2 + 17/2)", "incircle_radius_approx": 0.8397535982186035, "source_dataset": "advanced_geometry", "source_index": 805, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -7), B=(-10, -1), and C=(6, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.299", "metadata": {"A": ["3", "-7"], "B": ["-10", "-1"], "C": ["6", "7"], "incircle_radius_exact": "4*sqrt(5)*sqrt(-4*sqrt(5) + sqrt(205))/sqrt(4*sqrt(5) + sqrt(205))", "incircle_radius_approx": 4.298839322621755, "source_dataset": "advanced_geometry", "source_index": 806, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, -1), B=(0, -10), and C=(-5, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.333, 7.333)", "metadata": {"A": ["-9", "-1"], "B": ["0", "-10"], "C": ["-5", "-2"], "ortho": ["13/3", "22/3"], "orthocenter_approx": [4.333333333333333, 7.333333333333333], "source_dataset": "advanced_geometry", "source_index": 807, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, -8), B=(6, 10), and C=(4, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.055", "metadata": {"A": ["4", "-8"], "B": ["6", "10"], "C": ["4", "-7"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(293)/2 + sqrt(82))*sqrt(-sqrt(293)/2 + 1/2 + sqrt(82))*sqrt(-sqrt(82) + 1/2 + sqrt(293)/2)/sqrt(1/2 + sqrt(293)/2 + sqrt(82))", "incircle_radius_approx": 0.055205898195999234, "source_dataset": "advanced_geometry", "source_index": 808, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -1), B=(4, 3), and C=(3, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "145.30°", "metadata": {"A": ["2", "-1"], "B": ["4", "3"], "C": ["3", "10"], "angle_ABC_degrees": 145.30484646876604, "source_dataset": "advanced_geometry", "source_index": 809, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, 3), B=(5, 4), and C=(1, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "55.43°", "metadata": {"A": ["-9", "3"], "B": ["5", "4"], "C": ["1", "9"], "angle_ABC_degrees": 55.42580852588479, "source_dataset": "advanced_geometry", "source_index": 810, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-1, -1), B=(1, 0), and C=(-7, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.106", "metadata": {"A": ["-1", "-1"], "B": ["1", "0"], "C": ["-7", "-5"], "incircle_radius_exact": "sqrt(-sqrt(13) + sqrt(5)/2 + sqrt(89)/2)*sqrt(-sqrt(5)/2 + sqrt(13) + sqrt(89)/2)*sqrt(-sqrt(89)/2 + sqrt(5)/2 + sqrt(13))/sqrt(sqrt(5)/2 + sqrt(13) + sqrt(89)/2)", "incircle_radius_approx": 0.10592574202905865, "source_dataset": "advanced_geometry", "source_index": 811, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(0, 6), B=(7, 9), and C=(5, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.388", "metadata": {"A": ["0", "6"], "B": ["7", "9"], "C": ["5", "9"], "incircle_radius_exact": "sqrt(-1 + sqrt(34)/2 + sqrt(58)/2)*sqrt(-sqrt(34)/2 + 1 + sqrt(58)/2)*sqrt(-sqrt(58)/2 + 1 + sqrt(34)/2)/sqrt(1 + sqrt(34)/2 + sqrt(58)/2)", "incircle_radius_approx": 0.3884318520414211, "source_dataset": "advanced_geometry", "source_index": 812, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, -5), B=(10, 7), and C=(-7, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(30.667, -44.667)", "metadata": {"A": ["-2", "-5"], "B": ["10", "7"], "C": ["-7", "-7"], "ortho": ["92/3", "-134/3"], "orthocenter_approx": [30.666666666666668, -44.666666666666664], "source_dataset": "advanced_geometry", "source_index": 813, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-3, 7), B=(-9, 9), and C=(-6, -2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, 7.000)", "metadata": {"A": ["-3", "7"], "B": ["-9", "9"], "C": ["-6", "-2"], "ortho": ["-3", "7"], "orthocenter_approx": [-3.0, 7.0], "source_dataset": "advanced_geometry", "source_index": 814, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -4), B=(5, 4), and C=(-3, 4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.438", "metadata": {"A": ["3", "-4"], "B": ["5", "4"], "C": ["-3", "4"], "incircle_radius_exact": "sqrt(-1 + sqrt(17))*sqrt(1 + sqrt(17))*sqrt(9 - sqrt(17))/sqrt(sqrt(17) + 9)", "incircle_radius_approx": 2.4384471871911697, "source_dataset": "advanced_geometry", "source_index": 815, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, -8), B=(1, -5), and C=(3, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.96°", "metadata": {"A": ["6", "-8"], "B": ["1", "-5"], "C": ["3", "-5"], "angle_ABC_degrees": 30.96375653207352, "source_dataset": "advanced_geometry", "source_index": 816, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, -7), B=(-8, 10), and C=(9, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "25.43°", "metadata": {"A": ["-3", "-7"], "B": ["-8", "10"], "C": ["9", "-9"], "angle_ABC_degrees": 25.43062954610098, "source_dataset": "advanced_geometry", "source_index": 817, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -8), B=(-8, 7), and C=(1, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.526, 5.263)", "metadata": {"A": ["-6", "-8"], "B": ["-8", "7"], "C": ["1", "6"], "ortho": ["-86/19", "100/19"], "orthocenter_approx": [-4.526315789473684, 5.2631578947368425], "source_dataset": "advanced_geometry", "source_index": 818, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(4, -9), B=(-10, -6), and C=(-6, 4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.664", "metadata": {"A": ["4", "-9"], "B": ["-10", "-6"], "C": ["-6", "4"], "incircle_radius_exact": "sqrt(-sqrt(205)/2 + sqrt(29) + sqrt(269)/2)*sqrt(-sqrt(269)/2 + sqrt(29) + sqrt(205)/2)*sqrt(-sqrt(29) + sqrt(205)/2 + sqrt(269)/2)/sqrt(sqrt(29) + sqrt(205)/2 + sqrt(269)/2)", "incircle_radius_approx": 3.663588998120967, "source_dataset": "advanced_geometry", "source_index": 819, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-4, 10), B=(6, -8), and C=(-6, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "51.48°", "metadata": {"A": ["-4", "10"], "B": ["6", "-8"], "C": ["-6", "-6"], "angle_ABC_degrees": 51.48307369289724, "source_dataset": "advanced_geometry", "source_index": 820, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-2, 7), B=(3, 10), and C=(8, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "83.48°", "metadata": {"A": ["-2", "7"], "B": ["3", "10"], "C": ["8", "-1"], "angle_ABC_degrees": 83.48019824834302, "source_dataset": "advanced_geometry", "source_index": 821, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, -10), B=(1, -4), and C=(-6, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.306", "metadata": {"A": ["5", "-10"], "B": ["1", "-4"], "C": ["-6", "10"], "incircle_radius_exact": "sqrt(-sqrt(13) + 7*sqrt(5)/2 + sqrt(521)/2)*sqrt(-7*sqrt(5)/2 + sqrt(13) + sqrt(521)/2)*sqrt(-sqrt(521)/2 + sqrt(13) + 7*sqrt(5)/2)/sqrt(sqrt(13) + 7*sqrt(5)/2 + sqrt(521)/2)", "incircle_radius_approx": 0.3064194694039431, "source_dataset": "advanced_geometry", "source_index": 822, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 7), B=(-10, 4), and C=(10, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.883", "metadata": {"A": ["6", "7"], "B": ["-10", "4"], "C": ["10", "-10"], "incircle_radius_exact": "sqrt(-sqrt(149) + sqrt(265)/2 + sqrt(305)/2)*sqrt(-sqrt(265)/2 + sqrt(305)/2 + sqrt(149))*sqrt(-sqrt(305)/2 + sqrt(265)/2 + sqrt(149))/sqrt(sqrt(265)/2 + sqrt(305)/2 + sqrt(149))", "incircle_radius_approx": 4.883401815571663, "source_dataset": "advanced_geometry", "source_index": 823, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, -10), B=(-6, 1), and C=(-10, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.571, 3.714)", "metadata": {"A": ["-4", "-10"], "B": ["-6", "1"], "C": ["-10", "2"], "ortho": ["-4/7", "26/7"], "orthocenter_approx": [-0.5714285714285714, 3.7142857142857144], "source_dataset": "advanced_geometry", "source_index": 824, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, 4), B=(1, -4), and C=(9, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(12.692, 1.846)", "metadata": {"A": ["10", "4"], "B": ["1", "-4"], "C": ["9", "6"], "ortho": ["165/13", "24/13"], "orthocenter_approx": [12.692307692307692, 1.8461538461538463], "source_dataset": "advanced_geometry", "source_index": 825, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, 0), B=(7, 6), and C=(-10, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.767", "metadata": {"A": ["-6", "0"], "B": ["7", "6"], "C": ["-10", "-8"], "incircle_radius_exact": "sqrt(-sqrt(205)/2 + 2*sqrt(5) + sqrt(485)/2)*sqrt(-sqrt(485)/2 + 2*sqrt(5) + sqrt(205)/2)*sqrt(-2*sqrt(5) + sqrt(205)/2 + sqrt(485)/2)/sqrt(2*sqrt(5) + sqrt(205)/2 + sqrt(485)/2)", "incircle_radius_approx": 1.7665968481847796, "source_dataset": "advanced_geometry", "source_index": 826, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, -3), B=(9, -3), and C=(4, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.467", "metadata": {"A": ["2", "-3"], "B": ["9", "-3"], "C": ["4", "7"], "incircle_radius_exact": "sqrt(-7/2 + sqrt(26) + 5*sqrt(5)/2)*sqrt(-sqrt(26) + 7/2 + 5*sqrt(5)/2)*sqrt(-5*sqrt(5)/2 + 7/2 + sqrt(26))/sqrt(7/2 + sqrt(26) + 5*sqrt(5)/2)", "incircle_radius_approx": 2.4666666200505976, "source_dataset": "advanced_geometry", "source_index": 827, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 4), B=(-9, 1), and C=(-1, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.000, 9.000)", "metadata": {"A": ["4", "4"], "B": ["-9", "1"], "C": ["-1", "9"], "ortho": ["-1", "9"], "orthocenter_approx": [-1.0, 9.0], "source_dataset": "advanced_geometry", "source_index": 828, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -7), B=(4, -7), and C=(0, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.833", "metadata": {"A": ["7", "-7"], "B": ["4", "-7"], "C": ["0", "-2"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(41)/2 + sqrt(74)/2)*sqrt(-sqrt(41)/2 + 3/2 + sqrt(74)/2)*sqrt(-sqrt(74)/2 + 3/2 + sqrt(41)/2)/sqrt(3/2 + sqrt(41)/2 + sqrt(74)/2)", "incircle_radius_approx": 0.8330811178177788, "source_dataset": "advanced_geometry", "source_index": 829, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(5, -6), B=(2, 3), and C=(5, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "130.60°", "metadata": {"A": ["5", "-6"], "B": ["2", "3"], "C": ["5", "8"], "angle_ABC_degrees": 130.6012946450045, "source_dataset": "advanced_geometry", "source_index": 830, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(8, -1), B=(10, 1), and C=(1, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-80.000, 71.000)", "metadata": {"A": ["8", "-1"], "B": ["10", "1"], "C": ["1", "-10"], "ortho": ["-80", "71"], "orthocenter_approx": [-80.0, 71.0], "source_dataset": "advanced_geometry", "source_index": 831, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-5, 1), B=(1, -10), and C=(-8, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "37.43°", "metadata": {"A": ["-5", "1"], "B": ["1", "-10"], "C": ["-8", "-6"], "angle_ABC_degrees": 37.4270513594566, "source_dataset": "advanced_geometry", "source_index": 832, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, 0), B=(3, 9), and C=(-3, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "60.95°", "metadata": {"A": ["-2", "0"], "B": ["3", "9"], "C": ["-3", "9"], "angle_ABC_degrees": 60.94539590092286, "source_dataset": "advanced_geometry", "source_index": 833, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, -2), B=(4, 3), and C=(-9, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "20.58°", "metadata": {"A": ["-10", "-2"], "B": ["4", "3"], "C": ["-9", "-8"], "angle_ABC_degrees": 20.582534251220515, "source_dataset": "advanced_geometry", "source_index": 834, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, -5), B=(5, -2), and C=(9, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(62.400, -26.600)", "metadata": {"A": ["3", "-5"], "B": ["5", "-2"], "C": ["9", "9"], "ortho": ["312/5", "-133/5"], "orthocenter_approx": [62.4, -26.6], "source_dataset": "advanced_geometry", "source_index": 835, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, -1), B=(0, 2), and C=(3, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "52.74°", "metadata": {"A": ["8", "-1"], "B": ["0", "2"], "C": ["3", "-8"], "angle_ABC_degrees": 52.74471054642291, "source_dataset": "advanced_geometry", "source_index": 836, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(7, 1), B=(-6, -10), and C=(-4, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-22.333, 15.667)", "metadata": {"A": ["7", "1"], "B": ["-6", "-10"], "C": ["-4", "-6"], "ortho": ["-67/3", "47/3"], "orthocenter_approx": [-22.333333333333332, 15.666666666666666], "source_dataset": "advanced_geometry", "source_index": 837, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, -2), B=(9, -4), and C=(8, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(7.556, -9.778)", "metadata": {"A": ["-8", "-2"], "B": ["9", "-4"], "C": ["8", "-6"], "ortho": ["68/9", "-88/9"], "orthocenter_approx": [7.555555555555555, -9.777777777777779], "source_dataset": "advanced_geometry", "source_index": 838, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(0, -5), B=(4, 10), and C=(-6, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "58.37°", "metadata": {"A": ["0", "-5"], "B": ["4", "10"], "C": ["-6", "7"], "angle_ABC_degrees": 58.369338587868825, "source_dataset": "advanced_geometry", "source_index": 839, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(10, 0), B=(-6, 10), and C=(10, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(11.875, 10.000)", "metadata": {"A": ["10", "0"], "B": ["-6", "10"], "C": ["10", "7"], "ortho": ["95/8", "10"], "orthocenter_approx": [11.875, 10.0], "source_dataset": "advanced_geometry", "source_index": 840, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, -5), B=(0, 8), and C=(6, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "48.86°", "metadata": {"A": ["-7", "-5"], "B": ["0", "8"], "C": ["6", "-8"], "angle_ABC_degrees": 48.85680098558985, "source_dataset": "advanced_geometry", "source_index": 841, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, 5), B=(-5, 9), and C=(7, -1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "87.06°", "metadata": {"A": ["-8", "5"], "B": ["-5", "9"], "C": ["7", "-1"], "angle_ABC_degrees": 87.06432655357884, "source_dataset": "advanced_geometry", "source_index": 842, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(3, -9), B=(-1, 3), and C=(-8, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.784", "metadata": {"A": ["3", "-9"], "B": ["-1", "3"], "C": ["-8", "0"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + sqrt(58)/2 + sqrt(202)/2)*sqrt(-sqrt(202)/2 + sqrt(58)/2 + 2*sqrt(10))*sqrt(-sqrt(58)/2 + 2*sqrt(10) + sqrt(202)/2)/sqrt(sqrt(58)/2 + 2*sqrt(10) + sqrt(202)/2)", "incircle_radius_approx": 2.7844202515668095, "source_dataset": "advanced_geometry", "source_index": 843, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(6, 7), B=(3, -7), and C=(-3, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.176, -1.176)", "metadata": {"A": ["6", "7"], "B": ["3", "-7"], "C": ["-3", "-1"], "ortho": ["-37/17", "-20/17"], "orthocenter_approx": [-2.176470588235294, -1.1764705882352942], "source_dataset": "advanced_geometry", "source_index": 844, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, 5), B=(-4, -4), and C=(1, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.143", "metadata": {"A": ["2", "5"], "B": ["-4", "-4"], "C": ["1", "9"], "incircle_radius_exact": "sqrt(-3*sqrt(13)/2 + sqrt(17)/2 + sqrt(194)/2)*sqrt(-sqrt(17)/2 + 3*sqrt(13)/2 + sqrt(194)/2)*sqrt(-sqrt(194)/2 + sqrt(17)/2 + 3*sqrt(13)/2)/sqrt(sqrt(17)/2 + 3*sqrt(13)/2 + sqrt(194)/2)", "incircle_radius_approx": 1.1431284164666997, "source_dataset": "advanced_geometry", "source_index": 845, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, 8), B=(4, 5), and C=(2, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(38.000, 22.000)", "metadata": {"A": ["3", "8"], "B": ["4", "5"], "C": ["2", "10"], "ortho": ["38", "22"], "orthocenter_approx": [38.0, 22.0], "source_dataset": "advanced_geometry", "source_index": 846, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(9, -3), B=(9, -8), and C=(-5, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "43.03°", "metadata": {"A": ["9", "-3"], "B": ["9", "-8"], "C": ["-5", "7"], "angle_ABC_degrees": 43.02506598911803, "source_dataset": "advanced_geometry", "source_index": 847, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -1), B=(-2, 6), and C=(-10, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.881, 4.017)", "metadata": {"A": ["-1", "-1"], "B": ["-2", "6"], "C": ["-10", "3"], "ortho": ["-170/59", "237/59"], "orthocenter_approx": [-2.8813559322033897, 4.016949152542373], "source_dataset": "advanced_geometry", "source_index": 848, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, -2), B=(6, -3), and C=(-3, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "47.73°", "metadata": {"A": ["2", "-2"], "B": ["6", "-3"], "C": ["-3", "-9"], "angle_ABC_degrees": 47.72631099390627, "source_dataset": "advanced_geometry", "source_index": 849, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -3), B=(4, 3), and C=(-4, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(10.308, -15.923)", "metadata": {"A": ["-1", "-3"], "B": ["4", "3"], "C": ["-4", "-4"], "ortho": ["134/13", "-207/13"], "orthocenter_approx": [10.307692307692308, -15.923076923076923], "source_dataset": "advanced_geometry", "source_index": 850, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-2, -4), B=(1, 4), and C=(10, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "57.43°", "metadata": {"A": ["-2", "-4"], "B": ["1", "4"], "C": ["10", "-8"], "angle_ABC_degrees": 57.42594286542749, "source_dataset": "advanced_geometry", "source_index": 851, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, -1), B=(1, -4), and C=(-9, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.012", "metadata": {"A": ["5", "-1"], "B": ["1", "-4"], "C": ["-9", "6"], "incircle_radius_exact": "sqrt(-5/2 + 5*sqrt(2) + 7*sqrt(5)/2)*sqrt(-5*sqrt(2) + 5/2 + 7*sqrt(5)/2)*sqrt(-7*sqrt(5)/2 + 5/2 + 5*sqrt(2))/sqrt(5/2 + 5*sqrt(2) + 7*sqrt(5)/2)", "incircle_radius_approx": 2.0118057667618943, "source_dataset": "advanced_geometry", "source_index": 852, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-10, 3), B=(-3, 7), and C=(-3, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "60.26°", "metadata": {"A": ["-10", "3"], "B": ["-3", "7"], "C": ["-3", "-3"], "angle_ABC_degrees": 60.25511870305778, "source_dataset": "advanced_geometry", "source_index": 853, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -7), B=(-5, 0), and C=(10, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(6.701, -2.127)", "metadata": {"A": ["8", "-7"], "B": ["-5", "0"], "C": ["10", "4"], "ortho": ["1052/157", "-334/157"], "orthocenter_approx": [6.7006369426751595, -2.127388535031847], "source_dataset": "advanced_geometry", "source_index": 854, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, -2), B=(2, -2), and C=(2, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.000, -2.000)", "metadata": {"A": ["-9", "-2"], "B": ["2", "-2"], "C": ["2", "-8"], "ortho": ["2", "-2"], "orthocenter_approx": [2.0, -2.0], "source_dataset": "advanced_geometry", "source_index": 855, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-6, -8), B=(0, -10), and C=(5, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.925", "metadata": {"A": ["-6", "-8"], "B": ["0", "-10"], "C": ["5", "-3"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(74)/2 + sqrt(146)/2)*sqrt(-sqrt(146)/2 + sqrt(10) + sqrt(74)/2)*sqrt(-sqrt(74)/2 + sqrt(10) + sqrt(146)/2)/sqrt(sqrt(10) + sqrt(74)/2 + sqrt(146)/2)", "incircle_radius_approx": 1.925218118702094, "source_dataset": "advanced_geometry", "source_index": 856, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, 8), B=(-3, 0), and C=(-8, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.400, 0.000)", "metadata": {"A": ["-8", "8"], "B": ["-3", "0"], "C": ["-8", "-1"], "ortho": ["-32/5", "0"], "orthocenter_approx": [-6.4, 0.0], "source_dataset": "advanced_geometry", "source_index": 857, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-3, 1), B=(-7, -8), and C=(-3, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.60°", "metadata": {"A": ["-3", "1"], "B": ["-7", "-8"], "C": ["-3", "0"], "angle_ABC_degrees": 2.602562202499806, "source_dataset": "advanced_geometry", "source_index": 858, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -7), B=(8, -6), and C=(2, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.092", "metadata": {"A": ["-1", "-7"], "B": ["8", "-6"], "C": ["2", "5"], "incircle_radius_exact": "sqrt(-sqrt(157)/2 + sqrt(82)/2 + 3*sqrt(17)/2)*sqrt(-3*sqrt(17)/2 + sqrt(82)/2 + sqrt(157)/2)*sqrt(-sqrt(82)/2 + 3*sqrt(17)/2 + sqrt(157)/2)/sqrt(sqrt(82)/2 + 3*sqrt(17)/2 + sqrt(157)/2)", "incircle_radius_approx": 3.092358490207602, "source_dataset": "advanced_geometry", "source_index": 859, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, 10), B=(2, 5), and C=(-8, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.948", "metadata": {"A": ["10", "10"], "B": ["2", "5"], "C": ["-8", "8"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + sqrt(89)/2 + sqrt(82))*sqrt(-sqrt(82) + sqrt(89)/2 + sqrt(109)/2)*sqrt(-sqrt(89)/2 + sqrt(109)/2 + sqrt(82))/sqrt(sqrt(89)/2 + sqrt(109)/2 + sqrt(82))", "incircle_radius_approx": 1.9481344522686719, "source_dataset": "advanced_geometry", "source_index": 860, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -8), B=(-1, 4), and C=(-5, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(79.000, 84.000)", "metadata": {"A": ["10", "-8"], "B": ["-1", "4"], "C": ["-5", "7"], "ortho": ["79", "84"], "orthocenter_approx": [79.0, 84.0], "source_dataset": "advanced_geometry", "source_index": 861, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -1), B=(6, 2), and C=(9, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "29.74°", "metadata": {"A": ["9", "-1"], "B": ["6", "2"], "C": ["9", "-9"], "angle_ABC_degrees": 29.74488129694222, "source_dataset": "advanced_geometry", "source_index": 862, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(1, 1), B=(7, 5), and C=(9, -10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "63.90°", "metadata": {"A": ["1", "1"], "B": ["7", "5"], "C": ["9", "-10"], "angle_ABC_degrees": 63.904575842611656, "source_dataset": "advanced_geometry", "source_index": 863, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, 8), B=(1, -8), and C=(-2, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-39.333, -8.000)", "metadata": {"A": ["-2", "8"], "B": ["1", "-8"], "C": ["-2", "-1"], "ortho": ["-118/3", "-8"], "orthocenter_approx": [-39.333333333333336, -8.0], "source_dataset": "advanced_geometry", "source_index": 864, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(2, -9), B=(-6, -1), and C=(-1, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "95.19°", "metadata": {"A": ["2", "-9"], "B": ["-6", "-1"], "C": ["-1", "5"], "angle_ABC_degrees": 95.1944289077348, "source_dataset": "advanced_geometry", "source_index": 865, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-9, 2), B=(-5, -9), and C=(-1, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.87°", "metadata": {"A": ["-9", "2"], "B": ["-5", "-9"], "C": ["-1", "10"], "angle_ABC_degrees": 31.87176456152795, "source_dataset": "advanced_geometry", "source_index": 866, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(2, 10), B=(-4, -3), and C=(-10, 9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.800, 6.600)", "metadata": {"A": ["2", "10"], "B": ["-4", "-3"], "C": ["-10", "9"], "ortho": ["-24/5", "33/5"], "orthocenter_approx": [-4.8, 6.6], "source_dataset": "advanced_geometry", "source_index": 867, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 4), B=(10, -8), and C=(9, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "4.70°", "metadata": {"A": ["9", "4"], "B": ["10", "-8"], "C": ["9", "-2"], "angle_ABC_degrees": 4.698680517299439, "source_dataset": "advanced_geometry", "source_index": 868, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 2), B=(-6, -5), and C=(6, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.329", "metadata": {"A": ["2", "2"], "B": ["-6", "-5"], "C": ["6", "-3"], "incircle_radius_exact": "sqrt(-sqrt(113)/2 + sqrt(41)/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(41)/2 + sqrt(113)/2)*sqrt(-sqrt(41)/2 + sqrt(113)/2 + sqrt(37))/sqrt(sqrt(41)/2 + sqrt(113)/2 + sqrt(37))", "incircle_radius_approx": 2.3288632199392456, "source_dataset": "advanced_geometry", "source_index": 869, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -4), B=(7, 3), and C=(10, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "69.37°", "metadata": {"A": ["-1", "-4"], "B": ["7", "3"], "C": ["10", "-5"], "angle_ABC_degrees": 69.37012005387382, "source_dataset": "advanced_geometry", "source_index": 870, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-10, 5), B=(8, 0), and C=(0, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-107.500, -385.000)", "metadata": {"A": ["-10", "5"], "B": ["8", "0"], "C": ["0", "2"], "ortho": ["-215/2", "-385"], "orthocenter_approx": [-107.5, -385.0], "source_dataset": "advanced_geometry", "source_index": 871, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-1, -8), B=(8, -5), and C=(0, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.000, 0.000)", "metadata": {"A": ["-1", "-8"], "B": ["8", "-5"], "C": ["0", "-6"], "ortho": ["-2", "0"], "orthocenter_approx": [-2.0, 0.0], "source_dataset": "advanced_geometry", "source_index": 872, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, 0), B=(5, -9), and C=(-2, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-45.000, -25.667)", "metadata": {"A": ["-1", "0"], "B": ["5", "-9"], "C": ["-2", "3"], "ortho": ["-45", "-77/3"], "orthocenter_approx": [-45.0, -25.666666666666668], "source_dataset": "advanced_geometry", "source_index": 873, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(8, -4), B=(6, -7), and C=(7, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "29.60°", "metadata": {"A": ["8", "-4"], "B": ["6", "-7"], "C": ["7", "7"], "angle_ABC_degrees": 29.60445074600491, "source_dataset": "advanced_geometry", "source_index": 874, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, -8), B=(-9, -8), and C=(10, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "40.10°", "metadata": {"A": ["6", "-8"], "B": ["-9", "-8"], "C": ["10", "8"], "angle_ABC_degrees": 40.100907546212234, "source_dataset": "advanced_geometry", "source_index": 875, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, 4), B=(-8, -6), and C=(1, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "53.02°", "metadata": {"A": ["10", "4"], "B": ["-8", "-6"], "C": ["1", "-10"], "angle_ABC_degrees": 53.017093073655325, "source_dataset": "advanced_geometry", "source_index": 876, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -2), B=(-1, -3), and C=(7, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "8.13°", "metadata": {"A": ["6", "-2"], "B": ["-1", "-3"], "C": ["7", "-3"], "angle_ABC_degrees": 8.13010235415598, "source_dataset": "advanced_geometry", "source_index": 877, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, 3), B=(10, -6), and C=(0, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "36.28°", "metadata": {"A": ["0", "3"], "B": ["10", "-6"], "C": ["0", "-5"], "angle_ABC_degrees": 36.27661935831702, "source_dataset": "advanced_geometry", "source_index": 878, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(1, 2), B=(-8, 9), and C=(-5, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.405", "metadata": {"A": ["1", "2"], "B": ["-8", "9"], "C": ["-5", "-10"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + 3*sqrt(5) + sqrt(370)/2)*sqrt(-sqrt(370)/2 + sqrt(130)/2 + 3*sqrt(5))*sqrt(-3*sqrt(5) + sqrt(130)/2 + sqrt(370)/2)/sqrt(sqrt(130)/2 + 3*sqrt(5) + sqrt(370)/2)", "incircle_radius_approx": 3.404947229334776, "source_dataset": "advanced_geometry", "source_index": 879, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, 5), B=(-4, -4), and C=(-8, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.391, -13.435)", "metadata": {"A": ["6", "5"], "B": ["-4", "-4"], "C": ["-8", "-3"], "ortho": ["32/23", "-309/23"], "orthocenter_approx": [1.391304347826087, -13.434782608695652], "source_dataset": "advanced_geometry", "source_index": 880, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-6, 9), B=(1, 7), and C=(9, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.049", "metadata": {"A": ["-6", "9"], "B": ["1", "7"], "C": ["9", "-10"], "incircle_radius_exact": "sqrt(-sqrt(353)/2 + sqrt(53)/2 + sqrt(586)/2)*sqrt(-sqrt(53)/2 + sqrt(353)/2 + sqrt(586)/2)*sqrt(-sqrt(586)/2 + sqrt(53)/2 + sqrt(353)/2)/sqrt(sqrt(53)/2 + sqrt(353)/2 + sqrt(586)/2)", "incircle_radius_approx": 2.048697703895533, "source_dataset": "advanced_geometry", "source_index": 881, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(0, 4), B=(-8, 2), and C=(10, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "26.57°", "metadata": {"A": ["0", "4"], "B": ["-8", "2"], "C": ["10", "-2"], "angle_ABC_degrees": 26.56505117707799, "source_dataset": "advanced_geometry", "source_index": 882, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, -2), B=(2, -9), and C=(-1, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(20.000, 27.000)", "metadata": {"A": ["-9", "-2"], "B": ["2", "-9"], "C": ["-1", "-6"], "ortho": ["20", "27"], "orthocenter_approx": [20.0, 27.0], "source_dataset": "advanced_geometry", "source_index": 883, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(3, -6), B=(9, 1), and C=(-5, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "79.14°", "metadata": {"A": ["3", "-6"], "B": ["9", "1"], "C": ["-5", "9"], "angle_ABC_degrees": 79.14358665193775, "source_dataset": "advanced_geometry", "source_index": 884, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-6, -8), B=(-7, -4), and C=(-1, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "120.96°", "metadata": {"A": ["-6", "-8"], "B": ["-7", "-4"], "C": ["-1", "2"], "angle_ABC_degrees": 120.96375653207352, "source_dataset": "advanced_geometry", "source_index": 885, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(3, 7), B=(-6, -6), and C=(5, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.101", "metadata": {"A": ["3", "7"], "B": ["-6", "-6"], "C": ["5", "-3"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(26) + sqrt(130)/2)*sqrt(-sqrt(130)/2 + sqrt(26) + 5*sqrt(10)/2)*sqrt(-sqrt(26) + sqrt(130)/2 + 5*sqrt(10)/2)/sqrt(sqrt(26) + sqrt(130)/2 + 5*sqrt(10)/2)", "incircle_radius_approx": 3.1006772602193293, "source_dataset": "advanced_geometry", "source_index": 886, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(6, -8), B=(-1, -8), and C=(9, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(9.000, -10.500)", "metadata": {"A": ["6", "-8"], "B": ["-1", "-8"], "C": ["9", "4"], "ortho": ["9", "-21/2"], "orthocenter_approx": [9.0, -10.5], "source_dataset": "advanced_geometry", "source_index": 887, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(7, 4), B=(-3, -2), and C=(5, -4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "45.00°", "metadata": {"A": ["7", "4"], "B": ["-3", "-2"], "C": ["5", "-4"], "angle_ABC_degrees": 45.0, "source_dataset": "advanced_geometry", "source_index": 888, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 7), B=(8, 4), and C=(-3, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.232", "metadata": {"A": ["2", "7"], "B": ["8", "4"], "C": ["-3", "-1"], "incircle_radius_exact": "sqrt(-sqrt(146)/2 + 3*sqrt(5)/2 + sqrt(89)/2)*sqrt(-3*sqrt(5)/2 + sqrt(89)/2 + sqrt(146)/2)*sqrt(-sqrt(89)/2 + 3*sqrt(5)/2 + sqrt(146)/2)/sqrt(3*sqrt(5)/2 + sqrt(89)/2 + sqrt(146)/2)", "incircle_radius_approx": 2.2320455026513777, "source_dataset": "advanced_geometry", "source_index": 889, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, 6), B=(10, 6), and C=(3, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "59.74°", "metadata": {"A": ["-7", "6"], "B": ["10", "6"], "C": ["3", "-6"], "angle_ABC_degrees": 59.74356283647074, "source_dataset": "advanced_geometry", "source_index": 890, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, -5), B=(7, 6), and C=(-10, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "55.72°", "metadata": {"A": ["8", "-5"], "B": ["7", "6"], "C": ["-10", "-8"], "angle_ABC_degrees": 55.72196905939098, "source_dataset": "advanced_geometry", "source_index": 891, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-3, -10), B=(-10, -8), and C=(-2, -6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.527", "metadata": {"A": ["-3", "-10"], "B": ["-10", "-8"], "C": ["-2", "-6"], "incircle_radius_exact": "sqrt(-sqrt(17)/2 + sqrt(53)/2)*sqrt(sqrt(17)/2 + sqrt(53)/2)*sqrt(-sqrt(53)/2 + 3*sqrt(17)/2)/sqrt(sqrt(53)/2 + 3*sqrt(17)/2)", "incircle_radius_approx": 1.526762096271739, "source_dataset": "advanced_geometry", "source_index": 892, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, 2), B=(-9, 0), and C=(-5, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "47.56°", "metadata": {"A": ["4", "2"], "B": ["-9", "0"], "C": ["-5", "6"], "angle_ABC_degrees": 47.563770211465005, "source_dataset": "advanced_geometry", "source_index": 893, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, -5), B=(-9, 9), and C=(-4, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-4.344, -7.295)", "metadata": {"A": ["3", "-5"], "B": ["-9", "9"], "C": ["-4", "-7"], "ortho": ["-265/61", "-445/61"], "orthocenter_approx": [-4.344262295081967, -7.295081967213115], "source_dataset": "advanced_geometry", "source_index": 894, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, 10), B=(-8, 9), and C=(3, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.76°", "metadata": {"A": ["-6", "10"], "B": ["-8", "9"], "C": ["3", "8"], "angle_ABC_degrees": 31.7594800848128, "source_dataset": "advanced_geometry", "source_index": 895, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(5, 1), B=(7, 10), and C=(-5, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.522", "metadata": {"A": ["5", "1"], "B": ["7", "10"], "C": ["-5", "0"], "incircle_radius_exact": "sqrt(-sqrt(101)/2 + sqrt(85)/2 + sqrt(61))*sqrt(-sqrt(61) + sqrt(85)/2 + sqrt(101)/2)*sqrt(-sqrt(85)/2 + sqrt(101)/2 + sqrt(61))/sqrt(sqrt(85)/2 + sqrt(101)/2 + sqrt(61))", "incircle_radius_approx": 2.5222184928223332, "source_dataset": "advanced_geometry", "source_index": 896, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-6, -3), B=(-3, 0), and C=(-7, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-19.333, 2.333)", "metadata": {"A": ["-6", "-3"], "B": ["-3", "0"], "C": ["-7", "-10"], "ortho": ["-58/3", "7/3"], "orthocenter_approx": [-19.333333333333332, 2.3333333333333335], "source_dataset": "advanced_geometry", "source_index": 897, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, -10), B=(-7, -1), and C=(9, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.799", "metadata": {"A": ["-2", "-10"], "B": ["-7", "-1"], "C": ["9", "8"], "incircle_radius_exact": "sqrt(-sqrt(106)/2 + sqrt(337)/2 + sqrt(445)/2)*sqrt(-sqrt(337)/2 + sqrt(106)/2 + sqrt(445)/2)*sqrt(-sqrt(445)/2 + sqrt(106)/2 + sqrt(337)/2)/sqrt(sqrt(106)/2 + sqrt(337)/2 + sqrt(445)/2)", "incircle_radius_approx": 3.799131438041289, "source_dataset": "advanced_geometry", "source_index": 898, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(6, 7), B=(2, 3), and C=(9, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.915", "metadata": {"A": ["6", "7"], "B": ["2", "3"], "C": ["9", "0"], "incircle_radius_exact": "sqrt(-2*sqrt(2) + sqrt(58))*sqrt(16*sqrt(2) + 8*sqrt(58))/(2*sqrt(2) + sqrt(58))", "incircle_radius_approx": 1.9149383924470873, "source_dataset": "advanced_geometry", "source_index": 899, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, -4), B=(9, -2), and C=(5, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.713", "metadata": {"A": ["-4", "-4"], "B": ["9", "-2"], "C": ["5", "9"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(137)/2 + sqrt(173)/2)*sqrt(-sqrt(137)/2 + sqrt(173)/2 + 5*sqrt(10)/2)*sqrt(-sqrt(173)/2 + sqrt(137)/2 + 5*sqrt(10)/2)/sqrt(sqrt(137)/2 + sqrt(173)/2 + 5*sqrt(10)/2)", "incircle_radius_approx": 3.712898555406321, "source_dataset": "advanced_geometry", "source_index": 900, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-3, 7), B=(-6, -5), and C=(4, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "33.98°", "metadata": {"A": ["-3", "7"], "B": ["-6", "-5"], "C": ["4", "4"], "angle_ABC_degrees": 33.97654403625686, "source_dataset": "advanced_geometry", "source_index": 901, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 6), B=(7, 1), and C=(-9, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.000, 13.600)", "metadata": {"A": ["0", "6"], "B": ["7", "1"], "C": ["-9", "1"], "ortho": ["0", "68/5"], "orthocenter_approx": [0.0, 13.6], "source_dataset": "advanced_geometry", "source_index": 902, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(0, 10), B=(-4, -3), and C=(-4, -6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.363", "metadata": {"A": ["0", "10"], "B": ["-4", "-3"], "C": ["-4", "-6"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(185)/2 + 2*sqrt(17))*sqrt(-2*sqrt(17) + 3/2 + sqrt(185)/2)*sqrt(-sqrt(185)/2 + 3/2 + 2*sqrt(17))/sqrt(3/2 + sqrt(185)/2 + 2*sqrt(17))", "incircle_radius_approx": 0.3626046653361882, "source_dataset": "advanced_geometry", "source_index": 903, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -2), B=(5, -1), and C=(0, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(2.480, 7.400)", "metadata": {"A": ["10", "-2"], "B": ["5", "-1"], "C": ["0", "-5"], "ortho": ["62/25", "37/5"], "orthocenter_approx": [2.48, 7.4], "source_dataset": "advanced_geometry", "source_index": 904, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-2, 1), B=(6, -7), and C=(-5, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-86.000, -76.000)", "metadata": {"A": ["-2", "1"], "B": ["6", "-7"], "C": ["-5", "5"], "ortho": ["-86", "-76"], "orthocenter_approx": [-86.0, -76.0], "source_dataset": "advanced_geometry", "source_index": 905, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(1, -10), B=(9, -8), and C=(-8, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "57.30°", "metadata": {"A": ["1", "-10"], "B": ["9", "-8"], "C": ["-8", "8"], "angle_ABC_degrees": 57.30053887899809, "source_dataset": "advanced_geometry", "source_index": 906, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(1, 10), B=(0, -2), and C=(9, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.045", "metadata": {"A": ["1", "10"], "B": ["0", "-2"], "C": ["9", "-2"], "incircle_radius_exact": "sqrt(-9/2 + sqrt(145)/2 + 2*sqrt(13))*sqrt(-2*sqrt(13) + 9/2 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + 9/2 + 2*sqrt(13))/sqrt(9/2 + sqrt(145)/2 + 2*sqrt(13))", "incircle_radius_approx": 3.0453589568106265, "source_dataset": "advanced_geometry", "source_index": 907, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-4, 3), B=(-4, 4), and C=(-9, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-2.800, 1.000)", "metadata": {"A": ["-4", "3"], "B": ["-4", "4"], "C": ["-9", "1"], "ortho": ["-14/5", "1"], "orthocenter_approx": [-2.8, 1.0], "source_dataset": "advanced_geometry", "source_index": 908, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-2, -2), B=(-1, -7), and C=(3, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.043, -4.609)", "metadata": {"A": ["-2", "-2"], "B": ["-1", "-7"], "C": ["3", "-4"], "ortho": ["-1/23", "-106/23"], "orthocenter_approx": [-0.043478260869565216, -4.608695652173913], "source_dataset": "advanced_geometry", "source_index": 909, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, 4), B=(-4, -9), and C=(-5, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.000, -10.538)", "metadata": {"A": ["-9", "4"], "B": ["-4", "-9"], "C": ["-5", "-9"], "ortho": ["-9", "-137/13"], "orthocenter_approx": [-9.0, -10.538461538461538], "source_dataset": "advanced_geometry", "source_index": 910, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-3, 2), B=(-2, 3), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-14.000, 3.000)", "metadata": {"A": ["-3", "2"], "B": ["-2", "3"], "C": ["-3", "-8"], "ortho": ["-14", "3"], "orthocenter_approx": [-14.0, 3.0], "source_dataset": "advanced_geometry", "source_index": 911, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(8, -9), B=(-5, 7), and C=(-1, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-41.000, -33.500)", "metadata": {"A": ["8", "-9"], "B": ["-5", "7"], "C": ["-1", "-1"], "ortho": ["-41", "-67/2"], "orthocenter_approx": [-41.0, -33.5], "source_dataset": "advanced_geometry", "source_index": 912, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, 4), B=(-2, 9), and C=(2, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "30.07°", "metadata": {"A": ["3", "4"], "B": ["-2", "9"], "C": ["2", "-6"], "angle_ABC_degrees": 30.068582821862442, "source_dataset": "advanced_geometry", "source_index": 913, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(2, -6), B=(3, 4), and C=(-8, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "108.73°", "metadata": {"A": ["2", "-6"], "B": ["3", "4"], "C": ["-8", "9"], "angle_ABC_degrees": 108.73336164291689, "source_dataset": "advanced_geometry", "source_index": 914, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, -4), B=(-2, 1), and C=(3, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.389", "metadata": {"A": ["-5", "-4"], "B": ["-2", "1"], "C": ["3", "6"], "incircle_radius_exact": "sqrt(-5*sqrt(2)/2 + sqrt(34)/2 + sqrt(41))*sqrt(-sqrt(34)/2 + 5*sqrt(2)/2 + sqrt(41))*sqrt(-sqrt(41) + sqrt(34)/2 + 5*sqrt(2)/2)/sqrt(sqrt(34)/2 + 5*sqrt(2)/2 + sqrt(41))", "incircle_radius_approx": 0.3889799160865446, "source_dataset": "advanced_geometry", "source_index": 915, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(9, 5), B=(-3, 3), and C=(2, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.570", "metadata": {"A": ["9", "5"], "B": ["-3", "3"], "C": ["2", "5"], "incircle_radius_exact": "sqrt(-7/2 + sqrt(29)/2 + sqrt(37))*sqrt(-sqrt(29)/2 + 7/2 + sqrt(37))*sqrt(-sqrt(37) + sqrt(29)/2 + 7/2)/sqrt(sqrt(29)/2 + 7/2 + sqrt(37))", "incircle_radius_approx": 0.5702487414987629, "source_dataset": "advanced_geometry", "source_index": 916, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, -7), B=(10, 3), and C=(2, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.116", "metadata": {"A": ["-4", "-7"], "B": ["10", "3"], "C": ["2", "-3"], "incircle_radius_exact": "sqrt(-5 + sqrt(13) + sqrt(74))*sqrt(-sqrt(13) + 5 + sqrt(74))*sqrt(-sqrt(74) + sqrt(13) + 5)/sqrt(sqrt(13) + 5 + sqrt(74))", "incircle_radius_approx": 0.11622584547603143, "source_dataset": "advanced_geometry", "source_index": 917, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, -5), B=(-9, -1), and C=(6, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.779", "metadata": {"A": ["7", "-5"], "B": ["-9", "-1"], "C": ["6", "2"], "incircle_radius_exact": "sqrt(-2*sqrt(17) + 5*sqrt(2)/2 + 3*sqrt(26)/2)*sqrt(-5*sqrt(2)/2 + 3*sqrt(26)/2 + 2*sqrt(17))*sqrt(-3*sqrt(26)/2 + 5*sqrt(2)/2 + 2*sqrt(17))/sqrt(5*sqrt(2)/2 + 3*sqrt(26)/2 + 2*sqrt(17))", "incircle_radius_approx": 2.779168158500829, "source_dataset": "advanced_geometry", "source_index": 918, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(6, -9), B=(-7, 3), and C=(7, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "23.06°", "metadata": {"A": ["6", "-9"], "B": ["-7", "3"], "C": ["7", "-2"], "angle_ABC_degrees": 23.055565899308164, "source_dataset": "advanced_geometry", "source_index": 919, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(8, 10), B=(10, 4), and C=(0, -6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.086", "metadata": {"A": ["8", "10"], "B": ["10", "4"], "C": ["0", "-6"], "incircle_radius_exact": "sqrt(-sqrt(10) + 5*sqrt(2) + 4*sqrt(5))*sqrt(-5*sqrt(2) + sqrt(10) + 4*sqrt(5))*sqrt(-4*sqrt(5) + sqrt(10) + 5*sqrt(2))/sqrt(sqrt(10) + 5*sqrt(2) + 4*sqrt(5))", "incircle_radius_approx": 2.085764837371034, "source_dataset": "advanced_geometry", "source_index": 920, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, -4), B=(-5, -10), and C=(1, 10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "16.99°", "metadata": {"A": ["-1", "-4"], "B": ["-5", "-10"], "C": ["1", "10"], "angle_ABC_degrees": 16.990823291986164, "source_dataset": "advanced_geometry", "source_index": 921, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(3, 7), B=(7, -8), and C=(-3, -4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "53.27°", "metadata": {"A": ["3", "7"], "B": ["7", "-8"], "C": ["-3", "-4"], "angle_ABC_degrees": 53.26717333551064, "source_dataset": "advanced_geometry", "source_index": 922, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(5, 0), B=(4, -2), and C=(-2, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(4.250, -1.125)", "metadata": {"A": ["5", "0"], "B": ["4", "-2"], "C": ["-2", "2"], "ortho": ["17/4", "-9/8"], "orthocenter_approx": [4.25, -1.125], "source_dataset": "advanced_geometry", "source_index": 923, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(0, 8), B=(5, 0), and C=(4, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(68.000, 42.000)", "metadata": {"A": ["0", "8"], "B": ["5", "0"], "C": ["4", "2"], "ortho": ["68", "42"], "orthocenter_approx": [68.0, 42.0], "source_dataset": "advanced_geometry", "source_index": 924, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-7, -4), B=(3, 7), and C=(-8, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "42.53°", "metadata": {"A": ["-7", "-4"], "B": ["3", "7"], "C": ["-8", "6"], "angle_ABC_degrees": 42.53188208617146, "source_dataset": "advanced_geometry", "source_index": 925, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, 8), B=(10, -9), and C=(-4, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-6.887, 1.132)", "metadata": {"A": ["-1", "8"], "B": ["10", "-9"], "C": ["-4", "3"], "ortho": ["-365/53", "60/53"], "orthocenter_approx": [-6.886792452830188, 1.1320754716981132], "source_dataset": "advanced_geometry", "source_index": 926, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(7, -8), B=(-1, -8), and C=(2, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.492", "metadata": {"A": ["7", "-8"], "B": ["-1", "-8"], "C": ["2", "-9"], "incircle_radius_exact": "sqrt(-4 + sqrt(10)/2 + sqrt(26)/2)*sqrt(-sqrt(10)/2 + sqrt(26)/2 + 4)*sqrt(-sqrt(26)/2 + sqrt(10)/2 + 4)/sqrt(sqrt(10)/2 + sqrt(26)/2 + 4)", "incircle_radius_approx": 0.4919656725115759, "source_dataset": "advanced_geometry", "source_index": 927, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -7), B=(7, -6), and C=(-4, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.501", "metadata": {"A": ["-7", "-7"], "B": ["7", "-6"], "C": ["-4", "-10"], "incircle_radius_exact": "sqrt(-sqrt(137)/2 + 3*sqrt(2)/2 + sqrt(197)/2)*sqrt(-sqrt(197)/2 + 3*sqrt(2)/2 + sqrt(137)/2)*sqrt(-3*sqrt(2)/2 + sqrt(137)/2 + sqrt(197)/2)/sqrt(3*sqrt(2)/2 + sqrt(137)/2 + sqrt(197)/2)", "incircle_radius_approx": 1.5008500091313615, "source_dataset": "advanced_geometry", "source_index": 928, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-3, -2), B=(-5, -8), and C=(-8, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.464", "metadata": {"A": ["-3", "-2"], "B": ["-5", "-8"], "C": ["-8", "-5"], "incircle_radius_exact": "sqrt(-sqrt(10) + 3*sqrt(2)/2 + sqrt(34)/2)*sqrt(-3*sqrt(2)/2 + sqrt(34)/2 + sqrt(10))*sqrt(-sqrt(34)/2 + 3*sqrt(2)/2 + sqrt(10))/sqrt(3*sqrt(2)/2 + sqrt(34)/2 + sqrt(10))", "incircle_radius_approx": 1.463579920305012, "source_dataset": "advanced_geometry", "source_index": 929, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, 7), B=(3, -4), and C=(2, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "34.14°", "metadata": {"A": ["-7", "7"], "B": ["3", "-4"], "C": ["2", "3"], "angle_ABC_degrees": 34.14358665193776, "source_dataset": "advanced_geometry", "source_index": 930, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, -9), B=(3, -2), and C=(-2, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-65.000, -46.000)", "metadata": {"A": ["9", "-9"], "B": ["3", "-2"], "C": ["-2", "8"], "ortho": ["-65", "-46"], "orthocenter_approx": [-65.0, -46.0], "source_dataset": "advanced_geometry", "source_index": 931, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(6, 9), B=(9, 8), and C=(-10, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.912", "metadata": {"A": ["6", "9"], "B": ["9", "8"], "C": ["-10", "2"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(305)/2 + sqrt(397)/2)*sqrt(-sqrt(305)/2 + sqrt(10)/2 + sqrt(397)/2)*sqrt(-sqrt(397)/2 + sqrt(10)/2 + sqrt(305)/2)/sqrt(sqrt(10)/2 + sqrt(305)/2 + sqrt(397)/2)", "incircle_radius_approx": 0.9124225808701494, "source_dataset": "advanced_geometry", "source_index": 932, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-9, 10), B=(8, 7), and C=(-9, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.974", "metadata": {"A": ["-9", "10"], "B": ["8", "7"], "C": ["-9", "-1"], "incircle_radius_exact": "sqrt(-11/2 + sqrt(298)/2 + sqrt(353)/2)*sqrt(-sqrt(298)/2 + 11/2 + sqrt(353)/2)*sqrt(-sqrt(353)/2 + 11/2 + sqrt(298)/2)/sqrt(11/2 + sqrt(298)/2 + sqrt(353)/2)", "incircle_radius_approx": 3.9744132182592273, "source_dataset": "advanced_geometry", "source_index": 933, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -9), B=(-1, 7), and C=(8, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.414", "metadata": {"A": ["-4", "-9"], "B": ["-1", "7"], "C": ["8", "5"], "incircle_radius_exact": "sqrt(-sqrt(265)/2 + 3*sqrt(85)/2)*sqrt(-sqrt(85)/2 + sqrt(265)/2)*sqrt(sqrt(85)/2 + sqrt(265)/2)/sqrt(sqrt(265)/2 + 3*sqrt(85)/2)", "incircle_radius_approx": 3.4139438327336866, "source_dataset": "advanced_geometry", "source_index": 934, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, 10), B=(7, -1), and C=(0, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "17.62°", "metadata": {"A": ["-9", "10"], "B": ["7", "-1"], "C": ["0", "8"], "angle_ABC_degrees": 17.616493361233395, "source_dataset": "advanced_geometry", "source_index": 935, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-6, -8), B=(2, -3), and C=(7, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "161.81°", "metadata": {"A": ["-6", "-8"], "B": ["2", "-3"], "C": ["7", "3"], "angle_ABC_degrees": 161.8109543003487, "source_dataset": "advanced_geometry", "source_index": 936, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, -4), B=(-7, 1), and C=(-5, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.361", "metadata": {"A": ["-5", "-4"], "B": ["-7", "1"], "C": ["-5", "-7"], "incircle_radius_exact": "sqrt(-3/2 + sqrt(29)/2 + sqrt(17))*sqrt(-sqrt(17) + 3/2 + sqrt(29)/2)*sqrt(-sqrt(29)/2 + 3/2 + sqrt(17))/sqrt(3/2 + sqrt(29)/2 + sqrt(17))", "incircle_radius_approx": 0.3607638946375979, "source_dataset": "advanced_geometry", "source_index": 937, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -5), B=(-4, -2), and C=(-1, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.328", "metadata": {"A": ["-1", "-5"], "B": ["-4", "-2"], "C": ["-1", "3"], "incircle_radius_exact": "sqrt(-4 + 3*sqrt(2)/2 + sqrt(34)/2)*sqrt(-3*sqrt(2)/2 + sqrt(34)/2 + 4)*sqrt(-sqrt(34)/2 + 3*sqrt(2)/2 + 4)/sqrt(3*sqrt(2)/2 + sqrt(34)/2 + 4)", "incircle_radius_approx": 1.3279042277377273, "source_dataset": "advanced_geometry", "source_index": 938, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-9, -6), B=(1, 1), and C=(5, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(0.852, -1.074)", "metadata": {"A": ["-9", "-6"], "B": ["1", "1"], "C": ["5", "-7"], "ortho": ["23/27", "-29/27"], "orthocenter_approx": [0.8518518518518519, -1.074074074074074], "source_dataset": "advanced_geometry", "source_index": 939, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-7, 1), B=(-8, 5), and C=(-2, -8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "10.74°", "metadata": {"A": ["-7", "1"], "B": ["-8", "5"], "C": ["-2", "-8"], "angle_ABC_degrees": 10.738897100905442, "source_dataset": "advanced_geometry", "source_index": 940, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 6), B=(2, 7), and C=(8, 8), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "23.50°", "metadata": {"A": ["6", "6"], "B": ["2", "7"], "C": ["8", "8"], "angle_ABC_degrees": 23.498565675952094, "source_dataset": "advanced_geometry", "source_index": 941, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(10, -7), B=(5, -6), and C=(-8, 9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "142.22°", "metadata": {"A": ["10", "-7"], "B": ["5", "-6"], "C": ["-8", "9"], "angle_ABC_degrees": 142.22431569404534, "source_dataset": "advanced_geometry", "source_index": 942, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-4, 1), B=(-10, 5), and C=(3, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.299", "metadata": {"A": ["-4", "1"], "B": ["-10", "5"], "C": ["3", "8"], "incircle_radius_exact": "sqrt(-sqrt(13) + 7*sqrt(2)/2 + sqrt(178)/2)*sqrt(-sqrt(178)/2 + sqrt(13) + 7*sqrt(2)/2)*sqrt(-7*sqrt(2)/2 + sqrt(13) + sqrt(178)/2)/sqrt(sqrt(13) + 7*sqrt(2)/2 + sqrt(178)/2)", "incircle_radius_approx": 2.2986798494829834, "source_dataset": "advanced_geometry", "source_index": 943, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-8, 1), B=(-5, -8), and C=(8, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-7.000, 0.000)", "metadata": {"A": ["-8", "1"], "B": ["-5", "-8"], "C": ["8", "5"], "ortho": ["-7", "0"], "orthocenter_approx": [-7.0, 0.0], "source_dataset": "advanced_geometry", "source_index": 944, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, 10), B=(6, -9), and C=(-10, -10), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "93.58°", "metadata": {"A": ["6", "10"], "B": ["6", "-9"], "C": ["-10", "-10"], "angle_ABC_degrees": 93.57633437499736, "source_dataset": "advanced_geometry", "source_index": 945, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, -7), B=(8, -7), and C=(-3, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.042", "metadata": {"A": ["9", "-7"], "B": ["8", "-7"], "C": ["-3", "-8"], "incircle_radius_exact": "sqrt(-1/2 + sqrt(122)/2 + sqrt(145)/2)*sqrt(-sqrt(122)/2 + 1/2 + sqrt(145)/2)*sqrt(-sqrt(145)/2 + 1/2 + sqrt(122)/2)/sqrt(1/2 + sqrt(122)/2 + sqrt(145)/2)", "incircle_radius_approx": 0.041516247083002625, "source_dataset": "advanced_geometry", "source_index": 946, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-6, -1), B=(-2, 3), and C=(-9, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-36.000, 20.000)", "metadata": {"A": ["-6", "-1"], "B": ["-2", "3"], "C": ["-9", "-7"], "ortho": ["-36", "20"], "orthocenter_approx": [-36.0, 20.0], "source_dataset": "advanced_geometry", "source_index": 947, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-9, -2), B=(-1, -2), and C=(-2, 9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "84.81°", "metadata": {"A": ["-9", "-2"], "B": ["-1", "-2"], "C": ["-2", "9"], "angle_ABC_degrees": 84.80557109226518, "source_dataset": "advanced_geometry", "source_index": 948, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(4, 10), B=(-5, -1), and C=(-2, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.696, 0.478)", "metadata": {"A": ["4", "10"], "B": ["-5", "-1"], "C": ["-2", "-5"], "ortho": ["-200/23", "11/23"], "orthocenter_approx": [-8.695652173913043, 0.4782608695652174], "source_dataset": "advanced_geometry", "source_index": 949, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(1, -10), B=(0, -1), and C=(9, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.193, -0.133)", "metadata": {"A": ["1", "-10"], "B": ["0", "-1"], "C": ["9", "1"], "ortho": ["-99/83", "-11/83"], "orthocenter_approx": [-1.1927710843373494, -0.13253012048192772], "source_dataset": "advanced_geometry", "source_index": 950, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, -6), B=(7, 0), and C=(9, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(17.111, -12.278)", "metadata": {"A": ["-8", "-6"], "B": ["7", "0"], "C": ["9", "8"], "ortho": ["154/9", "-221/18"], "orthocenter_approx": [17.11111111111111, -12.277777777777779], "source_dataset": "advanced_geometry", "source_index": 951, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(7, 2), B=(10, 1), and C=(-9, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(72.000, 249.000)", "metadata": {"A": ["7", "2"], "B": ["10", "1"], "C": ["-9", "6"], "ortho": ["72", "249"], "orthocenter_approx": [72.0, 249.0], "source_dataset": "advanced_geometry", "source_index": 952, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, -3), B=(3, -1), and C=(-7, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.656, -2.688)", "metadata": {"A": ["-1", "-3"], "B": ["3", "-1"], "C": ["-7", "10"], "ortho": ["-21/32", "-43/16"], "orthocenter_approx": [-0.65625, -2.6875], "source_dataset": "advanced_geometry", "source_index": 953, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(5, -3), B=(-7, -7), and C=(1, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.284", "metadata": {"A": ["5", "-3"], "B": ["-7", "-7"], "C": ["1", "6"], "incircle_radius_exact": "sqrt(-2*sqrt(10) + sqrt(97)/2 + sqrt(233)/2)*sqrt(-sqrt(233)/2 + sqrt(97)/2 + 2*sqrt(10))*sqrt(-sqrt(97)/2 + 2*sqrt(10) + sqrt(233)/2)/sqrt(sqrt(97)/2 + 2*sqrt(10) + sqrt(233)/2)", "incircle_radius_approx": 3.2836977729886354, "source_dataset": "advanced_geometry", "source_index": 954, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, -1), B=(-2, -5), and C=(1, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.411", "metadata": {"A": ["4", "-1"], "B": ["-2", "-5"], "C": ["1", "-4"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + 3*sqrt(2)/2 + sqrt(13))*sqrt(-sqrt(13) + sqrt(10)/2 + 3*sqrt(2)/2)*sqrt(-3*sqrt(2)/2 + sqrt(10)/2 + sqrt(13))/sqrt(sqrt(10)/2 + 3*sqrt(2)/2 + sqrt(13))", "incircle_radius_approx": 0.41050844424644284, "source_dataset": "advanced_geometry", "source_index": 955, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 9), B=(7, 5), and C=(-5, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(9.364, 3.182)", "metadata": {"A": ["5", "9"], "B": ["7", "5"], "C": ["-5", "-4"], "ortho": ["103/11", "35/11"], "orthocenter_approx": [9.363636363636363, 3.1818181818181817], "source_dataset": "advanced_geometry", "source_index": 956, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-8, 8), B=(8, 7), and C=(5, 5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "37.27°", "metadata": {"A": ["-8", "8"], "B": ["8", "7"], "C": ["5", "5"], "angle_ABC_degrees": 37.26640190097714, "source_dataset": "advanced_geometry", "source_index": 957, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 2), B=(1, -8), and C=(4, -5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.795", "metadata": {"A": ["-7", "2"], "B": ["1", "-8"], "C": ["4", "-5"], "incircle_radius_exact": "sqrt(-sqrt(170)/2 + 3*sqrt(2)/2 + sqrt(41))*sqrt(-3*sqrt(2)/2 + sqrt(41) + sqrt(170)/2)*sqrt(-sqrt(41) + 3*sqrt(2)/2 + sqrt(170)/2)/sqrt(3*sqrt(2)/2 + sqrt(41) + sqrt(170)/2)", "incircle_radius_approx": 1.7947775579137595, "source_dataset": "advanced_geometry", "source_index": 958, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-4, -8), B=(7, 2), and C=(-5, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.244, 1.069)", "metadata": {"A": ["-4", "-8"], "B": ["7", "2"], "C": ["-5", "3"], "ortho": ["-425/131", "140/131"], "orthocenter_approx": [-3.2442748091603053, 1.0687022900763359], "source_dataset": "advanced_geometry", "source_index": 959, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, 3), B=(-3, -6), and C=(-6, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "90.00°", "metadata": {"A": ["9", "3"], "B": ["-3", "-6"], "C": ["-6", "-2"], "angle_ABC_degrees": 90.0, "source_dataset": "advanced_geometry", "source_index": 960, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(7, 1), B=(-3, 4), and C=(4, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.059", "metadata": {"A": ["7", "1"], "B": ["-3", "4"], "C": ["4", "7"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + 3*sqrt(5)/2 + sqrt(58)/2)*sqrt(-3*sqrt(5)/2 + sqrt(58)/2 + sqrt(109)/2)*sqrt(-sqrt(58)/2 + 3*sqrt(5)/2 + sqrt(109)/2)/sqrt(3*sqrt(5)/2 + sqrt(58)/2 + sqrt(109)/2)", "incircle_radius_approx": 2.0594175439254463, "source_dataset": "advanced_geometry", "source_index": 961, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-7, 9), B=(7, 6), and C=(10, -4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.838", "metadata": {"A": ["-7", "9"], "B": ["7", "6"], "C": ["10", "-4"], "incircle_radius_exact": "sqrt(-sqrt(109)/2 + sqrt(205)/2 + sqrt(458)/2)*sqrt(-sqrt(205)/2 + sqrt(109)/2 + sqrt(458)/2)*sqrt(-sqrt(458)/2 + sqrt(109)/2 + sqrt(205)/2)/sqrt(sqrt(109)/2 + sqrt(205)/2 + sqrt(458)/2)", "incircle_radius_approx": 2.8380126014605134, "source_dataset": "advanced_geometry", "source_index": 962, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(2, 7), B=(3, 8), and C=(10, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.663", "metadata": {"A": ["2", "7"], "B": ["3", "8"], "C": ["10", "-1"], "incircle_radius_exact": "sqrt(-sqrt(130)/2 + 9*sqrt(2)/2)*sqrt(-7*sqrt(2)/2 + sqrt(130)/2)*sqrt(7*sqrt(2)/2 + sqrt(130)/2)/sqrt(sqrt(130)/2 + 9*sqrt(2)/2)", "incircle_radius_approx": 0.6630839051832378, "source_dataset": "advanced_geometry", "source_index": 963, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-9, -3), B=(4, -10), and C=(-4, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.824", "metadata": {"A": ["-9", "-3"], "B": ["4", "-10"], "C": ["-4", "8"], "incircle_radius_exact": "sqrt(-sqrt(146)/2 + sqrt(218)/2 + sqrt(97))*sqrt(-sqrt(218)/2 + sqrt(146)/2 + sqrt(97))*sqrt(-sqrt(97) + sqrt(146)/2 + sqrt(218)/2)/sqrt(sqrt(146)/2 + sqrt(218)/2 + sqrt(97))", "incircle_radius_approx": 3.8242080615504275, "source_dataset": "advanced_geometry", "source_index": 964, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(9, -5), B=(-1, -5), and C=(-3, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, -7.182)", "metadata": {"A": ["9", "-5"], "B": ["-1", "-5"], "C": ["-3", "6"], "ortho": ["-3", "-79/11"], "orthocenter_approx": [-3.0, -7.181818181818182], "source_dataset": "advanced_geometry", "source_index": 965, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(4, 0), B=(-8, -7), and C=(-9, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "104.74°", "metadata": {"A": ["4", "0"], "B": ["-8", "-7"], "C": ["-9", "-6"], "angle_ABC_degrees": 104.74356283647073, "source_dataset": "advanced_geometry", "source_index": 966, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 5), B=(-8, 6), and C=(5, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.714, 12.429)", "metadata": {"A": ["-10", "5"], "B": ["-8", "6"], "C": ["5", "-5"], "ortho": ["-26/7", "87/7"], "orthocenter_approx": [-3.7142857142857144, 12.428571428571429], "source_dataset": "advanced_geometry", "source_index": 967, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(9, -3), B=(7, -8), and C=(3, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "41.78°", "metadata": {"A": ["9", "-3"], "B": ["7", "-8"], "C": ["3", "3"], "angle_ABC_degrees": 41.78451600825179, "source_dataset": "advanced_geometry", "source_index": 968, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-2, -10), B=(3, -5), and C=(4, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.089", "metadata": {"A": ["-2", "-10"], "B": ["3", "-5"], "C": ["4", "2"], "incircle_radius_exact": "3*sqrt(5)*sqrt(-3*sqrt(5) + 5*sqrt(2))/sqrt(3*sqrt(5) + 5*sqrt(2))", "incircle_radius_approx": 1.0885916380983185, "source_dataset": "advanced_geometry", "source_index": 969, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, -6), B=(2, -4), and C=(9, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.803", "metadata": {"A": ["-5", "-6"], "B": ["2", "-4"], "C": ["9", "-10"], "incircle_radius_exact": "sqrt(-sqrt(53)/2 + sqrt(85)/2)*sqrt(sqrt(53)/2 + sqrt(85)/2)*sqrt(-sqrt(85)/2 + 3*sqrt(53)/2)/sqrt(sqrt(85)/2 + 3*sqrt(53)/2)", "incircle_radius_approx": 1.8029693157926667, "source_dataset": "advanced_geometry", "source_index": 970, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-2, -5), B=(8, 8), and C=(-10, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.782", "metadata": {"A": ["-2", "-5"], "B": ["8", "8"], "C": ["-10", "2"], "incircle_radius_exact": "sqrt(-3*sqrt(10) + sqrt(113)/2 + sqrt(269)/2)*sqrt(-sqrt(113)/2 + sqrt(269)/2 + 3*sqrt(10))*sqrt(-sqrt(269)/2 + sqrt(113)/2 + 3*sqrt(10))/sqrt(sqrt(113)/2 + sqrt(269)/2 + 3*sqrt(10))", "incircle_radius_approx": 3.7821950188447353, "source_dataset": "advanced_geometry", "source_index": 971, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-10, -3), B=(-2, -9), and C=(3, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "75.75°", "metadata": {"A": ["-10", "-3"], "B": ["-2", "-9"], "C": ["3", "3"], "angle_ABC_degrees": 75.74996730219641, "source_dataset": "advanced_geometry", "source_index": 972, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-1, 5), B=(6, 2), and C=(-2, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-0.941, 5.471)", "metadata": {"A": ["-1", "5"], "B": ["6", "2"], "C": ["-2", "3"], "ortho": ["-16/17", "93/17"], "orthocenter_approx": [-0.9411764705882353, 5.470588235294118], "source_dataset": "advanced_geometry", "source_index": 973, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(2, 7), B=(1, 7), and C=(-5, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.196", "metadata": {"A": ["2", "7"], "B": ["1", "7"], "C": ["-5", "10"], "incircle_radius_exact": "sqrt(-1/2 + 3*sqrt(5)/2 + sqrt(58)/2)*sqrt(-3*sqrt(5)/2 + 1/2 + sqrt(58)/2)*sqrt(-sqrt(58)/2 + 1/2 + 3*sqrt(5)/2)/sqrt(1/2 + 3*sqrt(5)/2 + sqrt(58)/2)", "incircle_radius_approx": 0.19577163242215506, "source_dataset": "advanced_geometry", "source_index": 974, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, 3), B=(-4, -7), and C=(1, -1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.975", "metadata": {"A": ["10", "3"], "B": ["-4", "-7"], "C": ["1", "-1"], "incircle_radius_exact": "sqrt(-sqrt(61)/2 + sqrt(97)/2 + sqrt(74))*sqrt(-sqrt(74) + sqrt(61)/2 + sqrt(97)/2)*sqrt(-sqrt(97)/2 + sqrt(61)/2 + sqrt(74))/sqrt(sqrt(61)/2 + sqrt(97)/2 + sqrt(74))", "incircle_radius_approx": 0.9752247588600185, "source_dataset": "advanced_geometry", "source_index": 975, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(10, 2), B=(-7, -3), and C=(8, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.670", "metadata": {"A": ["10", "2"], "B": ["-7", "-3"], "C": ["8", "0"], "incircle_radius_exact": "sqrt(-sqrt(2) + 3*sqrt(26)/2 + sqrt(314)/2)*sqrt(-3*sqrt(26)/2 + sqrt(2) + sqrt(314)/2)*sqrt(-sqrt(314)/2 + sqrt(2) + 3*sqrt(26)/2)/sqrt(sqrt(2) + 3*sqrt(26)/2 + sqrt(314)/2)", "incircle_radius_approx": 0.6695395341121774, "source_dataset": "advanced_geometry", "source_index": 976, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(0, -4), B=(-7, -3), and C=(4, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.895, -5.737)", "metadata": {"A": ["0", "-4"], "B": ["-7", "-3"], "C": ["4", "9"], "ortho": ["36/19", "-109/19"], "orthocenter_approx": [1.894736842105263, -5.7368421052631575], "source_dataset": "advanced_geometry", "source_index": 977, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-10, 4), B=(2, 8), and C=(-5, -1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-5.500, 0.500)", "metadata": {"A": ["-10", "4"], "B": ["2", "8"], "C": ["-5", "-1"], "ortho": ["-11/2", "1/2"], "orthocenter_approx": [-5.5, 0.5], "source_dataset": "advanced_geometry", "source_index": 978, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(6, -5), B=(8, -3), and C=(-2, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "90.00°", "metadata": {"A": ["6", "-5"], "B": ["8", "-3"], "C": ["-2", "7"], "angle_ABC_degrees": 90.0, "source_dataset": "advanced_geometry", "source_index": 979, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(3, 9), B=(1, -8), and C=(2, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-38.538, 11.769)", "metadata": {"A": ["3", "9"], "B": ["1", "-8"], "C": ["2", "7"], "ortho": ["-501/13", "153/13"], "orthocenter_approx": [-38.53846153846154, 11.76923076923077], "source_dataset": "advanced_geometry", "source_index": 980, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(4, 5), B=(-6, 3), and C=(-1, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "3.505", "metadata": {"A": ["4", "5"], "B": ["-6", "3"], "C": ["-1", "-10"], "incircle_radius_exact": "sqrt(-5*sqrt(10)/2 + sqrt(26) + sqrt(194)/2)*sqrt(-sqrt(194)/2 + sqrt(26) + 5*sqrt(10)/2)*sqrt(-sqrt(26) + sqrt(194)/2 + 5*sqrt(10)/2)/sqrt(sqrt(26) + sqrt(194)/2 + 5*sqrt(10)/2)", "incircle_radius_approx": 3.5054496065561245, "source_dataset": "advanced_geometry", "source_index": 981, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, 2), B=(0, 6), and C=(-7, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(1.421, 14.526)", "metadata": {"A": ["5", "2"], "B": ["0", "6"], "C": ["-7", "4"], "ortho": ["27/19", "276/19"], "orthocenter_approx": [1.4210526315789473, 14.526315789473685], "source_dataset": "advanced_geometry", "source_index": 982, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-9, -9), B=(-4, 1), and C=(-3, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(47.571, -15.286)", "metadata": {"A": ["-9", "-9"], "B": ["-4", "1"], "C": ["-3", "10"], "ortho": ["333/7", "-107/7"], "orthocenter_approx": [47.57142857142857, -15.285714285714286], "source_dataset": "advanced_geometry", "source_index": 983, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(5, -2), B=(2, -2), and C=(-3, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.000, -6.000)", "metadata": {"A": ["5", "-2"], "B": ["2", "-2"], "C": ["-3", "8"], "ortho": ["-3", "-6"], "orthocenter_approx": [-3.0, -6.0], "source_dataset": "advanced_geometry", "source_index": 984, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(-7, -8), B=(7, 5), and C=(-3, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-1.375, -11.750)", "metadata": {"A": ["-7", "-8"], "B": ["7", "5"], "C": ["-3", "-10"], "ortho": ["-11/8", "-47/4"], "orthocenter_approx": [-1.375, -11.75], "source_dataset": "advanced_geometry", "source_index": 985, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(9, 2), B=(10, -2), and C=(-4, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.538", "metadata": {"A": ["9", "2"], "B": ["10", "-2"], "C": ["-4", "4"], "incircle_radius_exact": "sqrt(-sqrt(17)/2 + sqrt(173)/2 + sqrt(58))*sqrt(-sqrt(173)/2 + sqrt(17)/2 + sqrt(58))*sqrt(-sqrt(58) + sqrt(17)/2 + sqrt(173)/2)/sqrt(sqrt(17)/2 + sqrt(173)/2 + sqrt(58))", "incircle_radius_approx": 1.5381019408614265, "source_dataset": "advanced_geometry", "source_index": 986, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(-1, 9), B=(6, -5), and C=(7, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "31.33°", "metadata": {"A": ["-1", "9"], "B": ["6", "-5"], "C": ["7", "7"], "angle_ABC_degrees": 31.328692867804172, "source_dataset": "advanced_geometry", "source_index": 987, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(3, 9), B=(-5, 10), and C=(-4, 4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-3.511, 7.915)", "metadata": {"A": ["3", "9"], "B": ["-5", "10"], "C": ["-4", "4"], "ortho": ["-165/47", "372/47"], "orthocenter_approx": [-3.5106382978723403, 7.914893617021277], "source_dataset": "advanced_geometry", "source_index": 988, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(8, 7), B=(-9, -9), and C=(0, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.74°", "metadata": {"A": ["8", "7"], "B": ["-9", "-9"], "C": ["0", "0"], "angle_ABC_degrees": 1.7357045889283889, "source_dataset": "advanced_geometry", "source_index": 989, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(10, 7), B=(3, -5), and C=(0, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.347", "metadata": {"A": ["10", "7"], "B": ["3", "-5"], "C": ["0", "-4"], "incircle_radius_exact": "sqrt(-sqrt(10)/2 + sqrt(193)/2 + sqrt(221)/2)*sqrt(-sqrt(193)/2 + sqrt(10)/2 + sqrt(221)/2)*sqrt(-sqrt(221)/2 + sqrt(10)/2 + sqrt(193)/2)/sqrt(sqrt(10)/2 + sqrt(193)/2 + sqrt(221)/2)", "incircle_radius_approx": 1.347084438238927, "source_dataset": "advanced_geometry", "source_index": 990, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-4, -2), B=(4, -8), and C=(10, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "143.13°", "metadata": {"A": ["-4", "-2"], "B": ["4", "-8"], "C": ["10", "-8"], "angle_ABC_degrees": 143.13010235415598, "source_dataset": "advanced_geometry", "source_index": 991, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(10, -8), B=(-8, -7), and C=(2, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-8.000, -188.000)", "metadata": {"A": ["10", "-8"], "B": ["-8", "-7"], "C": ["2", "-8"], "ortho": ["-8", "-188"], "orthocenter_approx": [-8.0, -188.0], "source_dataset": "advanced_geometry", "source_index": 992, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given a triangle with vertices A=(4, -6), B=(-2, -5), and C=(-8, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "136.85°", "metadata": {"A": ["4", "-6"], "B": ["-2", "-5"], "C": ["-8", "-9"], "angle_ABC_degrees": 136.8476102659946, "source_dataset": "advanced_geometry", "source_index": 993, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-8, -8), B=(-9, -8), and C=(1, 4). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.380", "metadata": {"A": ["-8", "-8"], "B": ["-9", "-8"], "C": ["1", "4"], "incircle_radius_exact": "sqrt(-7 + sqrt(61))*sqrt(7 + sqrt(61))*sqrt(8 - sqrt(61))/sqrt(sqrt(61) + 8)", "incircle_radius_approx": 0.3795006481866912, "source_dataset": "advanced_geometry", "source_index": 994, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "For triangle with vertices A=(4, 5), B=(-7, -9), and C=(-10, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-9.816, 0.855)", "metadata": {"A": ["4", "5"], "B": ["-7", "-9"], "C": ["-10", "1"], "ortho": ["-373/38", "65/76"], "orthocenter_approx": [-9.81578947368421, 0.8552631578947368], "source_dataset": "advanced_geometry", "source_index": 995, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Given triangle ABC with coordinates A=(-8, -2), B=(-6, -7), and C=(1, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "(-11.458, 0.017)", "metadata": {"A": ["-8", "-2"], "B": ["-6", "-7"], "C": ["1", "5"], "ortho": ["-676/59", "1/59"], "orthocenter_approx": [-11.457627118644067, 0.01694915254237288], "source_dataset": "advanced_geometry", "source_index": 996, "task_type": "orthocenter", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "In triangle ABC with coordinates A=(-5, -4), B=(-7, -9), and C=(-3, 2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "1.82°", "metadata": {"A": ["-5", "-4"], "B": ["-7", "-9"], "C": ["-3", "2"], "angle_ABC_degrees": 1.8183029644518336, "source_dataset": "advanced_geometry", "source_index": 997, "task_type": "angle_measure", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 8), B=(-3, 1), and C=(-2, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "0.410", "metadata": {"A": ["-5", "8"], "B": ["-3", "1"], "C": ["-2", "-10"], "incircle_radius_exact": "sqrt(-sqrt(122)/2 + sqrt(53)/2 + 3*sqrt(37)/2)*sqrt(-3*sqrt(37)/2 + sqrt(53)/2 + sqrt(122)/2)*sqrt(-sqrt(53)/2 + sqrt(122)/2 + 3*sqrt(37)/2)/sqrt(sqrt(53)/2 + sqrt(122)/2 + 3*sqrt(37)/2)", "incircle_radius_approx": 0.41013012105610486, "source_dataset": "advanced_geometry", "source_index": 998, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}
{"question": "Consider triangle ABC with coordinates A=(-5, 4), B=(2, -10), and C=(-4, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n", "answer": "2.058", "metadata": {"A": ["-5", "4"], "B": ["2", "-10"], "C": ["-4", "-8"], "incircle_radius_exact": "sqrt(-sqrt(10) + sqrt(145)/2 + 7*sqrt(5)/2)*sqrt(-sqrt(145)/2 + sqrt(10) + 7*sqrt(5)/2)*sqrt(-7*sqrt(5)/2 + sqrt(10) + sqrt(145)/2)/sqrt(sqrt(10) + sqrt(145)/2 + 7*sqrt(5)/2)", "incircle_radius_approx": 2.057696290604211, "source_dataset": "advanced_geometry", "source_index": 999, "task_type": "incircle_radius", "difficulty": {"min_coord": -10, "max_coord": 10}}}