messages listlengths 1 1 | ground_truth stringlengths 5 21 | dataset stringclasses 1 value | type stringclasses 1 value |
|---|---|---|---|
[
{
"content": "Given triangle ABC with coordinates A=(-1, 8), B=(2, 6), 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"
}
] | (-1.600, 9.600) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-10, -8), B=(-8, -5), and C=(-5, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-10.000, -1.667) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-4, 3), B=(-10, -7), and C=(7, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-3.823, 1.494) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-1, -4), B=(-10, -8), 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"
}
] | 56.03° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-4, 4), B=(-6, -7), and C=(-9, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.882 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(5, -10), B=(1, -5), and C=(10, -6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 45.00° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(6, 8), B=(2, -10), and C=(-3, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 3.196 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-2, 9), B=(-8, -6), and C=(7, -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"
}
] | 56.89° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-6, -9), B=(-7, -9), and C=(-1, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.463 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -3), B=(9, -1), and C=(-1, 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"
}
] | 1.640 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(5, 4), B=(-4, -1), 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"
}
] | 1.770 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-4, 0), B=(-1, 9), and C=(-5, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.906 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(0, 1), B=(-7, 2), 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"
}
] | (1.143, 2.000) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-4, -7), B=(-3, 6), and C=(-6, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 40.60° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-10, -6), B=(-9, -3), and C=(-2, -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"
}
] | (-9.000, -3.667) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(6, 4), B=(5, -9), and C=(-5, 0), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (1.921, -0.532) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(1, 4), B=(-4, 2), and C=(7, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.655 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(2, -1), B=(-3, -10), and C=(0, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 19.05° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -6), B=(8, -5), and C=(2, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.575 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-5, 5), B=(-10, -10), and C=(-4, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.125 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -10), B=(-5, 8), and C=(0, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.903 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, 1), B=(10, -4), and C=(2, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.823 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-10, 0), B=(-6, 7), 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"
}
] | 3.197 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-8, 1), B=(-3, -9), 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"
}
] | 84.96° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(9, -3), B=(3, 2), and C=(5, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.939 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-1, 9), B=(6, -5), and C=(7, 7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 31.33° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-1, 10), B=(8, 10), and C=(7, 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"
}
] | 75.96° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(10, -9), B=(4, 2), and C=(3, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 36.74° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-3, -9), B=(8, 0), and C=(-1, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 57.72° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-8, 2), B=(-2, 2), and C=(-4, -8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-4.000, 1.200) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, -2), B=(-8, -6), 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"
}
] | 0.321 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, -6), B=(3, -1), and C=(-5, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (3.750, -7.000) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-5, -6), B=(-6, 6), and C=(-1, -7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.562 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-4, -1), B=(2, -9), and C=(0, -2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 20.92° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-6, -8), B=(-5, 10), and C=(1, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 20.71° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(0, -10), B=(-6, 1), and C=(-3, -7), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 8.05° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(2, 4), B=(-10, 4), and C=(9, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.216 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-3, 5), B=(-2, 2), and C=(7, 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"
}
] | 1.344 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, -1), B=(7, 0), and C=(-10, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-38.000, 135.000) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(2, -7), B=(7, -10), and C=(9, -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"
}
] | 104.04° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(5, -8), B=(-2, 2), and C=(6, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.489 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(9, 1), B=(-5, 7), and C=(3, 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"
}
] | 1.093 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(2, 3), B=(3, -9), 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"
}
] | 63.80° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-9, 7), B=(9, 9), and C=(2, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-9.000, 108.000) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(0, 7), B=(10, 4), and C=(2, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (5.740, 3.468) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-1, -3), B=(2, 10), and C=(-2, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-4.122, 9.490) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(4, 4), B=(9, -4), and C=(5, -4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 57.99° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-4, 5), B=(3, -1), 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"
}
] | 17.98° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(8, -7), B=(-3, -8), 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"
}
] | 1.634 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(1, -4), B=(-8, -5), and C=(-5, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-3.914, -2.771) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(2, 5), B=(-7, -7), 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"
}
] | 67.17° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-5, 0), B=(-10, 8), and C=(2, -4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-27.333, -22.333) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-7, -8), B=(7, 5), and C=(-3, -10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-1.375, -11.750) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(7, 6), B=(0, -7), and C=(-4, -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"
}
] | (34.581, -30.774) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-2, -6), B=(3, 0), and C=(-3, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.646 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(0, -6), B=(7, 7), and C=(-1, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 54.57° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-2, -3), B=(-7, 4), and C=(8, -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"
}
] | 1.484 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(3, 7), B=(8, 5), and C=(2, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 83.19° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, 6), B=(-8, 9), and C=(1, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.032 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(8, -3), B=(8, 10), and C=(2, -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"
}
] | (19.333, -7.000) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(10, 7), B=(3, -5), and C=(0, -4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.347 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(5, 8), B=(-8, 4), and C=(-7, 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"
}
] | (-8.000, 11.250) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-3, -1), B=(-5, 10), and C=(-1, 6). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.493 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(3, -10), B=(-9, -10), and C=(-6, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-6.000, -1.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-3, 4), B=(4, 7), and C=(-9, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.503 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-8, 0), B=(-7, 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"
}
] | (7.000, 52.500) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(3, 8), B=(9, 10), 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"
}
] | 7.13° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, 6), B=(-5, -7), and C=(8, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 4.166 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(10, -2), B=(-3, 7), and C=(3, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (10.000, 17.111) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -2), B=(-2, -3), 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"
}
] | 0.700 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, 0), B=(6, -3), 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"
}
] | (3.333, -19.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 5), B=(3, 9), 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"
}
] | 1.064 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, -8), B=(3, 6), and C=(-3, 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"
}
] | (-6.231, 0.231) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(10, -8), B=(-5, -7), and C=(3, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 3.31° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(10, 2), B=(-7, -3), and C=(8, 0). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.670 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(5, -5), B=(-3, 3), and C=(-9, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 81.87° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-7, -7), B=(-2, -10), and C=(-5, -9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-11.000, -19.000) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(9, -2), B=(9, 10), and C=(0, 7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 71.57° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-1, 5), B=(6, 2), and C=(-2, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-0.941, 5.471) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(5, 3), B=(10, 8), 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"
}
] | 23.20° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(9, -5), B=(0, -4), and C=(-1, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.387 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(10, -6), B=(5, -9), and C=(-7, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (2.921, -12.535) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(6, -1), B=(2, -9), 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"
}
] | 61.07° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-9, -5), B=(5, 4), and C=(-8, 2), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 23.99° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, 6), B=(10, -10), 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"
}
] | 0.929 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 3), B=(-8, -5), and C=(10, 1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 3.251 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(8, 2), B=(10, 0), and C=(10, 10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 45.00° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-9, -2), B=(2, -9), and C=(-1, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (20.000, 27.000) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(5, 3), B=(8, -4), 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"
}
] | 66.80° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 7), B=(7, 3), and C=(-7, -8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 4.124 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(5, -7), B=(-6, 6), and C=(2, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (15.000, 13.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(5, 6), B=(-6, -9), and C=(-8, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.601 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(4, 6), B=(8, 0), and C=(-8, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.717 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-3, -9), B=(6, -9), and C=(0, 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"
}
] | 71.57° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(9, -2), B=(-1, 5), and C=(6, -3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 13.82° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(6, 2), B=(-6, 6), 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"
}
] | 2.307 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-6, -3), B=(9, 2), and C=(8, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (8.000, -5.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(8, -4), B=(5, -1), and C=(-3, 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"
}
] | (13.333, 17.333) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-3, 7), B=(4, 8), and C=(5, 7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.485 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(10, -10), B=(-2, 9), and C=(10, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 4.264 | Puzzle | advanced_geometry |
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