messages listlengths 1 1 | ground_truth stringlengths 5 21 | dataset stringclasses 1 value | type stringclasses 1 value |
|---|---|---|---|
[
{
"content": "Given a triangle with vertices A=(-9, 9), B=(2, 5), and C=(-3, 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"
}
] | 10.98° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(8, 9), B=(-9, 5), 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",
"role": "user"
}
] | 58.24° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-10, 4), B=(-5, -4), 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"
}
] | 103.57° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 25.18° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, -2), B=(-6, 4), 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",
"role": "user"
}
] | 2.143 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 57.30° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, -5), B=(-7, -7), and C=(-6, -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"
}
] | (-7.000, -0.500) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, 1), B=(7, 5), and C=(-7, -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"
}
] | (43.000, -55.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-10, -4), B=(-5, 8), 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",
"role": "user"
}
] | (-6.672, 11.530) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(3, 0), B=(10, 7), 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",
"role": "user"
}
] | 1.97° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(1, -1), B=(-5, 4), and C=(-10, 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"
}
] | (6.053, 24.263) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(1, -2), B=(10, 6), and C=(9, -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",
"role": "user"
}
] | 41.24° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 2.432 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(0, -8), B=(-8, 1), 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",
"role": "user"
}
] | (-1.571, -3.286) | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 37.27° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(8, -10), B=(-5, -7), 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",
"role": "user"
}
] | 60.12° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(1, -7), B=(8, 4), 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"
}
] | (-86.000, 51.000) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(0, 9), B=(1, -1), 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",
"role": "user"
}
] | 167.95° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(0, 0), B=(-5, 9), 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"
}
] | (-21.465, -19.814) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 4), B=(5, -3), 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"
}
] | 0.867 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, 5), B=(10, -3), 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",
"role": "user"
}
] | 2.321 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, 2), B=(10, 6), and C=(-2, -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"
}
] | (-23.818, 34.182) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(6, 8), B=(5, 8), 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",
"role": "user"
}
] | 56.31° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-9, -3), B=(7, 4), 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",
"role": "user"
}
] | 2.631 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, 3), B=(8, 8), 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",
"role": "user"
}
] | (5.333, -5.333) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-5, 6), B=(-10, -6), 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",
"role": "user"
}
] | 2.610 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-6, 6), B=(2, -7), 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",
"role": "user"
}
] | 52.68° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(3, -6), B=(10, -5), 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",
"role": "user"
}
] | 0.181 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (-6.000, 4.286) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -3), B=(-9, 0), and C=(8, 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"
}
] | 1.540 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-7, -7), B=(7, -2), 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"
}
] | (9.111, -23.111) | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 0.082 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(4, 1), B=(0, 3), 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",
"role": "user"
}
] | 156.37° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, -9), B=(7, 10), 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",
"role": "user"
}
] | (24.907, -5.349) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-5, -5), B=(4, 4), 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",
"role": "user"
}
] | 32.47° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(9, -8), B=(0, -8), and C=(-6, -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"
}
] | 153.43° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(5, 7), B=(8, 6), 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",
"role": "user"
}
] | 52.13° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-2, -6), B=(-5, 2), 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",
"role": "user"
}
] | (1.207, 6.828) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-6, -8), B=(-1, -4), 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",
"role": "user"
}
] | 1.347 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(8, -6), B=(-6, -5), 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",
"role": "user"
}
] | (9.141, -8.028) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(2, 10), B=(-5, -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",
"role": "user"
}
] | 17.56° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(5, -10), B=(-5, 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",
"role": "user"
}
] | 3.638 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-7, -7), B=(3, 4), and C=(2, -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"
}
] | 37.87° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(5, 10), B=(1, 2), and C=(7, -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"
}
] | 124.82° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (68.000, 42.000) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-10, 7), B=(4, -5), 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",
"role": "user"
}
] | 2.28° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 0.971 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 2.370 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(2, 10), B=(0, 0), 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",
"role": "user"
}
] | 1.571 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (3.800, -2.400) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, 1), B=(8, 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",
"role": "user"
}
] | (6.444, 0.111) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(6, -3), B=(-7, -7), and C=(0, 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"
}
] | (2.128, -0.915) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, -6), B=(7, 5), 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",
"role": "user"
}
] | (8.935, -0.032) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, 5), B=(2, -7), and C=(0, -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"
}
] | (-14.000, -3.000) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(9, 9), B=(1, -3), 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",
"role": "user"
}
] | 22.62° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(2, 8), B=(0, -4), 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",
"role": "user"
}
] | 125.54° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(0, -4), B=(5, 10), 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",
"role": "user"
}
] | (5.000, -6.143) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-7, 2), B=(3, -7), 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",
"role": "user"
}
] | 87.82° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(5, 2), B=(6, -1), and C=(1, -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",
"role": "user"
}
] | (6.294, -1.235) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(10, 7), B=(1, 5), 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",
"role": "user"
}
] | 167.47° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 48.76° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(9, 2), B=(-2, 1), 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",
"role": "user"
}
] | 83.88° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, -8), B=(3, -2), 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",
"role": "user"
}
] | 0.397 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(5, -7), B=(-10, -4), 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",
"role": "user"
}
] | 1.797 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(9, 2), B=(-8, -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",
"role": "user"
}
] | (-11.129, 35.548) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-10, 2), B=(-3, -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",
"role": "user"
}
] | (-27.000, -15.000) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(1, 6), B=(9, -8), 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",
"role": "user"
}
] | (1.000, -4.000) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, 8), B=(-4, -4), 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",
"role": "user"
}
] | (0.588, -3.294) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-8, -1), B=(4, -4), 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"
}
] | (4.000, -25.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-2, -3), B=(10, 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",
"role": "user"
}
] | 1.909 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 99.92° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 0.306 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (15.000, 0.000) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-5, -7), B=(-8, 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",
"role": "user"
}
] | 40.75° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(1, -9), B=(7, -8), and C=(1, 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"
}
] | 2.565 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(3, -6), B=(0, 9), 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",
"role": "user"
}
] | 1.545 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-9, -6), B=(2, -5), 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",
"role": "user"
}
] | 1.818 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-3, -6), B=(2, -6), 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",
"role": "user"
}
] | (4.000, -6.933) | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (-8.500, 0.000) | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 38.16° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-9, 2), B=(9, -8), 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",
"role": "user"
}
] | (-9.000, -42.200) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(1, 1), B=(7, 4), 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",
"role": "user"
}
] | (1.556, 0.889) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-3, -3), B=(-5, 5), 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",
"role": "user"
}
] | 85.43° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (11.870, 10.087) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-4, -1), B=(6, -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"
}
] | (-3.509, -6.887) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(6, -5), B=(-4, 5), and C=(0, -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"
}
] | 0.700 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-6, 6), B=(2, 1), and C=(0, 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",
"role": "user"
}
] | 12.99° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | (-20.143, -11.286) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-7, -7), B=(-8, -9), 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",
"role": "user"
}
] | (-61.000, 44.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, -10), B=(-3, -6), and C=(-7, 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"
}
] | (-23.400, -19.600) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, -6), B=(9, -5), 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"
}
] | 0.960 | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 1.111 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(5, 7), B=(-8, -4), 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",
"role": "user"
}
] | 43.81° | Puzzle | advanced_geometry |
[
{
"content": "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",
"role": "user"
}
] | 3.179 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-3, -10), B=(1, -8), 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",
"role": "user"
}
] | (5.500, -8.000) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 8), B=(10, -2), 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",
"role": "user"
}
] | 2.033 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(9, -6), B=(-1, 1), 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",
"role": "user"
}
] | 0.608 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(5, 7), B=(3, -10), 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",
"role": "user"
}
] | (8.321, 5.962) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(3, 0), B=(5, 0), 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",
"role": "user"
}
] | 23.96° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(3, 5), B=(6, 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",
"role": "user"
}
] | (26.222, -20.333) | Puzzle | advanced_geometry |
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