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, -8), B=(-6, -8), 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"
}
] | 75.96° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-6, -3), B=(-4, -7), and C=(-7, -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"
}
] | 18.43° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(7, -4), B=(-7, 10), and C=(3, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.432 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(0, -10), B=(-5, -1), 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"
}
] | (-1.630, -12.795) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(9, 3), B=(-9, -5), and C=(1, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.254 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-7, -10), B=(10, 7), and C=(-5, 3), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 30.07° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(3, 4), B=(0, 4), and C=(4, -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"
}
] | (4.000, 4.800) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-3, 9), B=(-5, 10), 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"
}
] | 49.40° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-10, -2), B=(-7, 6), 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"
}
] | 104.85° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-8, -5), B=(-6, 10), and C=(6, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 71.03° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(7, 9), B=(2, 10), and C=(-4, 6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-1.000, 21.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-5, -8), B=(-9, 5), and C=(-1, 9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.999 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-10, 10), B=(7, 5), and C=(-2, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (7.000, 40.600) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(0, -3), B=(-5, -3), and C=(8, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (8.000, -107.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-9, 1), B=(-9, -3), and C=(7, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-9.000, 1.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(4, -8), B=(0, -10), and C=(-8, -3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.566 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-8, -10), B=(1, -1), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (8.333, -19.333) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(0, -9), B=(-3, 3), 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"
}
] | 3.175 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(6, 10), B=(-8, -10), and C=(-7, -1), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 28.65° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(5, -9), B=(-5, 7), and C=(-8, -5), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 46.04° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(5, 4), B=(-3, 10), 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"
}
] | (-3.000, -9.333) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-2, 2), B=(-6, -8), and C=(-2, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (30.500, -8.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-4, -2), B=(-8, -2), and C=(6, 2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (6.000, -37.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, 5), B=(-5, -3), and C=(10, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-3.000, 0.250) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-7, -10), B=(10, 3), and C=(1, 7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (0.719, 7.368) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(7, -2), B=(5, -10), 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"
}
] | (11.545, -5.636) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(0, 4), B=(-4, -8), 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"
}
] | (-7.500, -3.500) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-3, 9), B=(-1, 8), and C=(-5, 1), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-1.444, 8.111) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(2, 10), B=(-2, 4), and C=(-9, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-302.000, 187.333) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(4, 7), B=(-1, -4), and C=(3, -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"
}
] | (10.118, -5.235) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(3, -2), B=(8, -10), 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"
}
] | 57.99° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(4, -2), B=(10, 0), and C=(5, -6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 31.76° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-2, 7), B=(1, 1), and C=(5, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.825 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-2, -9), B=(10, 9), 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"
}
] | 56.31° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(8, -9), B=(-7, 10), and C=(1, 1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (164.176, 129.824) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-9, -4), B=(4, 10), and C=(-3, -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"
}
] | 10.41° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(7, -7), B=(0, 8), 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"
}
] | 14.40° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-4, 10), B=(-3, 3), 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"
}
] | 53.13° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-5, -4), B=(-9, -5), and C=(5, 8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (12.789, -23.158) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(7, 9), B=(-10, 0), 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"
}
] | 15.37° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-5, -10), B=(-2, 5), and C=(0, -2), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (13.529, -4.706) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-6, -9), B=(0, -10), 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"
}
] | 80.54° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(7, 7), B=(2, -10), and C=(7, -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"
}
] | 15.62° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(0, 2), B=(8, 10), and C=(0, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-17.000, 10.000) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(6, 3), B=(10, -9), and C=(6, 2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.158 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(6, -7), B=(0, 3), and C=(7, 0), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 35.84° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(8, -2), B=(-9, -6), and C=(-4, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 3.844 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(1, 5), B=(10, 9), 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"
}
] | (10.000, -33.250) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(1, 8), B=(1, 2), and C=(3, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.371 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, 6), B=(-5, 5), and C=(7, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-0.412, 8.824) | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-5, 7), B=(4, -3), and C=(-2, -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"
}
] | 29.58° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(3, 4), B=(-8, -10), and C=(-3, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 22.63° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(6, -10), B=(-9, 9), and C=(-6, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-21.696, -16.391) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-6, -3), B=(8, 6), and C=(-7, -4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.95° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(6, 1), B=(-10, -9), and C=(-10, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-7.500, 1.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-5, 9), B=(2, -5), 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"
}
] | (5.091, -4.455) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(5, -6), B=(2, -4), and C=(-8, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-394.000, -576.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(10, -6), B=(-4, -3), and C=(2, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (0.636, -15.364) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(8, 0), B=(10, -7), and C=(2, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.883 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(2, 8), B=(3, 10), 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"
}
] | (158.000, -83.000) | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-2, 8), B=(1, -8), and C=(-2, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-39.333, -8.000) | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(8, 3), B=(-4, 10), and C=(2, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (8.517, 3.172) | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(1, -8), B=(7, 6), and C=(-10, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 25.38° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(6, -10), B=(4, -10), and C=(0, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.924 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-7, 1), B=(-2, 4), 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"
}
] | 0.775 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-2, 1), B=(-3, 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"
}
] | 4.97° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-9, 1), B=(3, 1), and C=(-6, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 18.43° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(2, 1), B=(-6, -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"
}
] | 0.649 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 10), B=(4, 8), and C=(-6, 5). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.898 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-7, -6), B=(10, -1), and C=(9, -9), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 66.49° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -6), B=(4, 10), and C=(-5, -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"
}
] | 3.321 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-1, 2), B=(4, -6), and C=(4, 3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (2.400, 2.000) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(4, 3), B=(4, 5), and C=(7, 0). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.497 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-7, 1), B=(5, 10), and C=(10, -2), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 75.75° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-10, -2), B=(4, 3), and C=(-9, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 20.58° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-5, 0), B=(1, -2), and C=(4, 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"
}
] | 98.13° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(6, 5), B=(10, 8), and C=(-6, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 19.52° | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(7, -3), B=(-10, 7), and C=(1, -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"
}
] | 8.82° | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-3, -4), B=(2, 9), and C=(-5, 10), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-1.208, 8.542) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(1, 0), B=(3, -3), and C=(10, 3). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.479 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-9, 5), B=(-6, 8), 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"
}
] | 125.54° | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(-10, -1), B=(7, -6), and C=(-4, 0), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 12.22° | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-1, -4), B=(-5, 4), and C=(9, -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"
}
] | 2.560 | Puzzle | advanced_geometry |
[
{
"content": "In triangle ABC with coordinates A=(-7, 3), B=(4, -5), and C=(-2, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 8.97° | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(-1, -1), B=(-10, 7), and C=(-3, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (6.215, 2.367) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-10, -5), B=(5, 4), 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"
}
] | 3.298 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(10, 4), B=(-7, 9), and C=(5, 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"
}
] | (1.102, -11.254) | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-5, 0), B=(-3, -9), 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"
}
] | 0.744 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, 9), B=(-3, 3), and C=(9, 4). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.131 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(0, 1), B=(-2, -4), and C=(1, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.183 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(7, -4), B=(6, -1), and C=(0, -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"
}
] | 1.260 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-5, -4), B=(-7, 1), and C=(-5, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.361 | Puzzle | advanced_geometry |
[
{
"content": "Given triangle ABC with coordinates A=(6, -6), B=(1, 8), and C=(9, -8), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (-15.500, -16.750) | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(-6, -9), B=(5, 10), and C=(9, -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"
}
] | 4.066 | Puzzle | advanced_geometry |
[
{
"content": "Given a triangle with vertices A=(7, 4), B=(3, 7), 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"
}
] | 74.93° | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -3), B=(3, -2), and C=(8, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 0.054 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(6, -4), B=(-9, 8), and C=(8, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 1.293 | Puzzle | advanced_geometry |
[
{
"content": "Consider triangle ABC with coordinates A=(3, -7), B=(-3, -10), and C=(-4, 1). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | 2.431 | Puzzle | advanced_geometry |
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(3, 9), B=(7, 6), 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"
}
] | 2.026 | Puzzle | advanced_geometry |
[
{
"content": "For triangle with vertices A=(-2, 1), B=(6, -9), and C=(10, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] | (0.511, 0.409) | Puzzle | advanced_geometry |
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