TTTXXX01/advanced_geometry · Datasets at Hugging Face

messages listlengths 1 1	ground_truth stringlengths 5 21	dataset stringclasses 1 value	type stringclasses 1 value
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 2), B=(-5, -2), 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", "role": "user" } ]	1.373	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(3, -1), B=(6, 6), 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", "role": "user" } ]	1.642	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-4, 1), B=(-8, -1), and C=(7, -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", "role": "user" } ]	(-1.000, 6.000)	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(8, -10), B=(7, -2), 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", "role": "user" } ]	(-47.000, 1.000)	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(6, 6), B=(-9, 0), 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", "role": "user" } ]	2.172	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	90.00°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	0.352	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	0.833	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(14.222, -6.444)	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	0.230	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(6, 1), B=(-3, 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", "role": "user" } ]	1.479	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(4, -2), B=(5, 1), and C=(0, 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", "role": "user" } ]	1.078	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(6, 2), B=(-7, 0), 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", "role": "user" } ]	2.621	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-7, 0), B=(10, 4), 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", "role": "user" } ]	23.63°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -7), B=(-1, 3), 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", "role": "user" } ]	0.840	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(0, 6), B=(4, 9), 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", "role": "user" } ]	1.056	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(7, -6), B=(6, 5), 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", "role": "user" } ]	0.892	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-4, 9), B=(6, -7), 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", "role": "user" } ]	65.70°	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-3, 3), B=(1, -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", "role": "user" } ]	(1.000, 1.857)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(4, -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", "role": "user" } ]	40.10°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(1, 1), B=(-8, 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", "role": "user" } ]	2.717	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-5, -1), B=(6, 0), 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", "role": "user" } ]	19.25°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	7.13°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -6), B=(3, -1), 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", "role": "user" } ]	1.014	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, 1), B=(3, -3), 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", "role": "user" } ]	1.516	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(1, 8), B=(5, -1), 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", "role": "user" } ]	39.47°	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(3, -1), B=(4, 1), 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", "role": "user" } ]	0.688	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(5, 8), B=(-4, 7), 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", "role": "user" } ]	(-3.291, 6.618)	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	145.30°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	60.95°	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-3, 9), B=(-7, 9), 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", "role": "user" } ]	56.31°	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(8, -1), B=(10, 0), 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", "role": "user" } ]	0.328	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(6, -6), B=(3, 2), 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", "role": "user" } ]	1.677	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(10, -6), B=(-6, -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", "role": "user" } ]	3.923	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(10, -3), B=(-4, 6), 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", "role": "user" } ]	1.536	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(8, -10), B=(-1, 9), and C=(1, -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", "role": "user" } ]	(-10.113, -12.264)	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	0.410	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	14.74°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(0.273, 7.455)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-7, 4), B=(-10, -10), 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", "role": "user" } ]	3.43°	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(-3, 2), B=(-2, 4), 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", "role": "user" } ]	0.513	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(4, 8), B=(1, -10), 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", "role": "user" } ]	40.73°	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-1, 10), B=(-9, 0), and C=(-10, -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", "role": "user" } ]	167.91°	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(-4, -9), B=(-10, 1), 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", "role": "user" } ]	(-10.000, -10.800)	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(7, -5), B=(3, -3), 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", "role": "user" } ]	(39.000, 99.000)	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(-4.000, 14.000)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(10, -3), B=(-4, 4), and C=(10, 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", "role": "user" } ]	26.57°	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(5, 4), B=(-9, 1), 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", "role": "user" } ]	(5.414, 3.402)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-3, -8), B=(-8, -9), and C=(-9, 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", "role": "user" } ]	83.09°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	3.974	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(6.477, -0.890)	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(-9, -7), B=(-9, -9), 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", "role": "user" } ]	0.876	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-1, -8), B=(3, 5), 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", "role": "user" } ]	72.90°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	54.52°	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(3, 9), B=(10, -2), 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", "role": "user" } ]	54.52°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(9, -9), B=(6, -6), 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", "role": "user" } ]	1.295	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	3.407	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(0, 2), B=(-1, 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", "role": "user" } ]	(4.091, 10.182)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(10, 3), B=(-5, 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", "role": "user" } ]	176.50°	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(5, 5), B=(-2, 6), 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", "role": "user" } ]	34.70°	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(10, 4), B=(6, 8), 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", "role": "user" } ]	49.76°	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-10, 9), B=(0, 3), and C=(-6, 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", "role": "user" } ]	(-12.692, -7.154)	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -4), B=(-1, 7), 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", "role": "user" } ]	3.912	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-9, 10), B=(1, -5), 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", "role": "user" } ]	2.73°	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(4, -8), B=(-9, -3), and C=(-7, 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", "role": "user" } ]	102.29°	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(10, -2), B=(-10, -1), 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", "role": "user" } ]	(1.578, -16.438)	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(8, 6), B=(8, 0), 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", "role": "user" } ]	(8.000, 6.000)	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-9, -1), B=(6, -1), and C=(-3, 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", "role": "user" } ]	29.05°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	0.223	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(7, 1), B=(-3, 0), 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", "role": "user" } ]	(-9.111, 17.111)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(0, -6), B=(-4, 6), 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", "role": "user" } ]	1.22°	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-3, 4), B=(6, -1), 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", "role": "user" } ]	(-2.204, -2.367)	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(-1, -2), B=(0, 10), and C=(0, 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", "role": "user" } ]	(36.000, -2.000)	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -7), B=(6, 3), 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", "role": "user" } ]	2.985	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(6, -3), B=(-5, -9), and C=(-4, 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", "role": "user" } ]	3.627	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(-5, 3), B=(9, 0), 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", "role": "user" } ]	0.381	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-4, -7), B=(9, 10), 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", "role": "user" } ]	(-14.305, 6.116)	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(4, 8), B=(-3, 2), 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", "role": "user" } ]	(-1.049, 2.390)	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(7, 10), B=(-7, -1), 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", "role": "user" } ]	(-12.500, 10.000)	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(9, 5), B=(-4, -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", "role": "user" } ]	81.09°	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(-2, 1), B=(-4, 10), 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", "role": "user" } ]	1.074	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	2.485	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-4, 1), B=(2, 2), 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", "role": "user" } ]	50.65°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -9), B=(-10, 0), 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", "role": "user" } ]	4.440	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(4, 9), B=(-5, 3), and C=(2, -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", "role": "user" } ]	3.503	Puzzle	advanced_geometry
[ { "content": "Given a triangle with vertices A=(-2, 6), B=(0, -1), and C=(3, 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", "role": "user" } ]	39.14°	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(8, 4), B=(0, 4), 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", "role": "user" } ]	(9.000, 5.125)	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 6), B=(-2, 9), 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", "role": "user" } ]	1.764	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(-10, 5), B=(10, 8), 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", "role": "user" } ]	(2.510, -1.736)	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(-0.571, 3.714)	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(2, -3), B=(-5, 7), 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", "role": "user" } ]	0.995	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	(-6.400, 0.000)	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(10, -9), B=(7, -7), and C=(2, 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", "role": "user" } ]	150.26°	Puzzle	advanced_geometry
[ { "content": "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", "role": "user" } ]	19.44°	Puzzle	advanced_geometry
[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -2), B=(-4, 8), 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", "role": "user" } ]	0.128	Puzzle	advanced_geometry
[ { "content": "Consider triangle ABC with coordinates A=(0, 7), B=(-10, -2), 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", "role": "user" } ]	1.221	Puzzle	advanced_geometry
[ { "content": "In triangle ABC with coordinates A=(-9, -10), B=(10, -3), and C=(0, -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", "role": "user" } ]	14.51°	Puzzle	advanced_geometry
[ { "content": "For triangle with vertices A=(-3, -6), B=(-7, -1), 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", "role": "user" } ]	(-7.222, -1.778)	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(4, 8), B=(-1, -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", "role": "user" } ]	(12.400, -6.000)	Puzzle	advanced_geometry
[ { "content": "Given triangle ABC with coordinates A=(5, 4), B=(-6, 1), and C=(-6, 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", "role": "user" } ]	(-6.273, 4.000)	Puzzle	advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-10, 2), B=(-5, -2), 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", "role": "user" } ]

1.373

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(3, -1), B=(6, 6), 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", "role": "user" } ]

1.642

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-4, 1), B=(-8, -1), and C=(7, -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", "role": "user" } ]

(-1.000, 6.000)

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(8, -10), B=(7, -2), 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", "role": "user" } ]

(-47.000, 1.000)

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(6, 6), B=(-9, 0), 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", "role": "user" } ]

2.172

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

90.00°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

0.352

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

0.833

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(14.222, -6.444)

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

0.230

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(6, 1), B=(-3, 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", "role": "user" } ]

1.479

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(4, -2), B=(5, 1), and C=(0, 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", "role": "user" } ]

1.078

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(6, 2), B=(-7, 0), 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", "role": "user" } ]

2.621

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-7, 0), B=(10, 4), 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", "role": "user" } ]

23.63°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -7), B=(-1, 3), 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", "role": "user" } ]

0.840

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(0, 6), B=(4, 9), 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", "role": "user" } ]

1.056

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(7, -6), B=(6, 5), 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", "role": "user" } ]

0.892

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-4, 9), B=(6, -7), 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", "role": "user" } ]

65.70°

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-3, 3), B=(1, -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", "role": "user" } ]

(1.000, 1.857)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(4, -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", "role": "user" } ]

40.10°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(1, 1), B=(-8, 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", "role": "user" } ]

2.717

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-5, -1), B=(6, 0), 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", "role": "user" } ]

19.25°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

7.13°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -6), B=(3, -1), 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", "role": "user" } ]

1.014

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, 1), B=(3, -3), 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", "role": "user" } ]

1.516

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(1, 8), B=(5, -1), 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", "role": "user" } ]

39.47°

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(3, -1), B=(4, 1), 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", "role": "user" } ]

0.688

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(5, 8), B=(-4, 7), 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", "role": "user" } ]

(-3.291, 6.618)

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

145.30°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

60.95°

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-3, 9), B=(-7, 9), 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", "role": "user" } ]

56.31°

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(8, -1), B=(10, 0), 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", "role": "user" } ]

0.328

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(6, -6), B=(3, 2), 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", "role": "user" } ]

1.677

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(10, -6), B=(-6, -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", "role": "user" } ]

3.923

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(10, -3), B=(-4, 6), 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", "role": "user" } ]

1.536

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(8, -10), B=(-1, 9), and C=(1, -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", "role": "user" } ]

(-10.113, -12.264)

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

0.410

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

14.74°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(0.273, 7.455)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-7, 4), B=(-10, -10), 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", "role": "user" } ]

3.43°

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(-3, 2), B=(-2, 4), 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", "role": "user" } ]

0.513

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(4, 8), B=(1, -10), 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", "role": "user" } ]

40.73°

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-1, 10), B=(-9, 0), and C=(-10, -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", "role": "user" } ]

167.91°

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(-4, -9), B=(-10, 1), 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", "role": "user" } ]

(-10.000, -10.800)

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(7, -5), B=(3, -3), 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", "role": "user" } ]

(39.000, 99.000)

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(-4.000, 14.000)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(10, -3), B=(-4, 4), and C=(10, 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", "role": "user" } ]

26.57°

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(5, 4), B=(-9, 1), 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", "role": "user" } ]

(5.414, 3.402)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-3, -8), B=(-8, -9), and C=(-9, 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", "role": "user" } ]

83.09°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

3.974

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(6.477, -0.890)

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(-9, -7), B=(-9, -9), 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", "role": "user" } ]

0.876

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-1, -8), B=(3, 5), 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", "role": "user" } ]

72.90°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

54.52°

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(3, 9), B=(10, -2), 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", "role": "user" } ]

54.52°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(9, -9), B=(6, -6), 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", "role": "user" } ]

1.295

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

3.407

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(0, 2), B=(-1, 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", "role": "user" } ]

(4.091, 10.182)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(10, 3), B=(-5, 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", "role": "user" } ]

176.50°

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(5, 5), B=(-2, 6), 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", "role": "user" } ]

34.70°

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(10, 4), B=(6, 8), 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", "role": "user" } ]

49.76°

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-10, 9), B=(0, 3), and C=(-6, 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", "role": "user" } ]

(-12.692, -7.154)

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -4), B=(-1, 7), 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", "role": "user" } ]

3.912

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-9, 10), B=(1, -5), 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", "role": "user" } ]

2.73°

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(4, -8), B=(-9, -3), and C=(-7, 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", "role": "user" } ]

102.29°

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(10, -2), B=(-10, -1), 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", "role": "user" } ]

(1.578, -16.438)

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(8, 6), B=(8, 0), 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", "role": "user" } ]

(8.000, 6.000)

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-9, -1), B=(6, -1), and C=(-3, 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", "role": "user" } ]

29.05°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

0.223

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(7, 1), B=(-3, 0), 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", "role": "user" } ]

(-9.111, 17.111)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(0, -6), B=(-4, 6), 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", "role": "user" } ]

1.22°

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-3, 4), B=(6, -1), 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", "role": "user" } ]

(-2.204, -2.367)

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(-1, -2), B=(0, 10), and C=(0, 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", "role": "user" } ]

(36.000, -2.000)

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -7), B=(6, 3), 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", "role": "user" } ]

2.985

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(6, -3), B=(-5, -9), and C=(-4, 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", "role": "user" } ]

3.627

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(-5, 3), B=(9, 0), 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", "role": "user" } ]

0.381

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-4, -7), B=(9, 10), 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", "role": "user" } ]

(-14.305, 6.116)

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(4, 8), B=(-3, 2), 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", "role": "user" } ]

(-1.049, 2.390)

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(7, 10), B=(-7, -1), 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", "role": "user" } ]

(-12.500, 10.000)

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(9, 5), B=(-4, -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", "role": "user" } ]

81.09°

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(-2, 1), B=(-4, 10), 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", "role": "user" } ]

1.074

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

2.485

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-4, 1), B=(2, 2), 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", "role": "user" } ]

50.65°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -9), B=(-10, 0), 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", "role": "user" } ]

4.440

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(4, 9), B=(-5, 3), and C=(2, -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", "role": "user" } ]

3.503

Puzzle

advanced_geometry

[ { "content": "Given a triangle with vertices A=(-2, 6), B=(0, -1), and C=(3, 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", "role": "user" } ]

39.14°

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(8, 4), B=(0, 4), 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", "role": "user" } ]

(9.000, 5.125)

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 6), B=(-2, 9), 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", "role": "user" } ]

1.764

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(-10, 5), B=(10, 8), 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", "role": "user" } ]

(2.510, -1.736)

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(-0.571, 3.714)

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(2, -3), B=(-5, 7), 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", "role": "user" } ]

0.995

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

(-6.400, 0.000)

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(10, -9), B=(7, -7), and C=(2, 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", "role": "user" } ]

150.26°

Puzzle

advanced_geometry

[ { "content": "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", "role": "user" } ]

19.44°

Puzzle

advanced_geometry

[ { "content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -2), B=(-4, 8), 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", "role": "user" } ]

0.128

Puzzle

advanced_geometry

[ { "content": "Consider triangle ABC with coordinates A=(0, 7), B=(-10, -2), 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", "role": "user" } ]

1.221

Puzzle

advanced_geometry

[ { "content": "In triangle ABC with coordinates A=(-9, -10), B=(10, -3), and C=(0, -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", "role": "user" } ]

14.51°

Puzzle

advanced_geometry

[ { "content": "For triangle with vertices A=(-3, -6), B=(-7, -1), 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", "role": "user" } ]

(-7.222, -1.778)

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(4, 8), B=(-1, -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", "role": "user" } ]

(12.400, -6.000)

Puzzle

advanced_geometry

[ { "content": "Given triangle ABC with coordinates A=(5, 4), B=(-6, 1), and C=(-6, 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", "role": "user" } ]

(-6.273, 4.000)

Puzzle

advanced_geometry