messages
listlengths 1
1
| ground_truth
stringlengths 5
21
| dataset
stringclasses 1
value | type
stringclasses 1
value |
|---|---|---|---|
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 10), B=(7, -6), and C=(-8, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
4.473
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(4, -5), B=(-10, 1), and C=(-9, -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"
}
] |
(-10.688, -7.938)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-5, -4), B=(-2, 1), and C=(3, 6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.389
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(4, 2), B=(-4, 7), 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"
}
] |
7.28°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(4, -8), B=(-6, 2), and C=(-9, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.629
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(6, -6), B=(-9, 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"
}
] |
30.53°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(1, 0), B=(4, -10), and C=(3, 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"
}
] |
10.99°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 3), B=(7, 3), and C=(-4, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
4.033
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(9, -3), B=(6, 3), and C=(-2, -10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
58.17°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(2, 3), B=(9, 10), and C=(-4, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.299
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -7), B=(7, -7), and C=(0, 7). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.737
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 3), B=(0, -8), and C=(0, 1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.498
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-7, 9), B=(5, 5), 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"
}
] |
(19.000, 35.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(9, -10), B=(-5, 1), and C=(4, 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"
}
] |
62.12°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(2, 8), B=(0, -9), and C=(-2, -9). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.929
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(6, -10), B=(-8, -6), and C=(-7, 5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.476
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-1, -7), B=(4, -7), 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"
}
] |
0.807
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-8, 7), B=(1, -4), and C=(-1, 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"
}
] |
28.98°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -3), B=(-10, 1), 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"
}
] |
2.059
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(5, 0), B=(-1, 1), and C=(-7, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-11.250, -19.500)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 3), B=(10, 8), and C=(-4, -8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
4.416
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(3, -5), B=(9, -3), 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"
}
] |
1.310
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(1, 6), B=(7, -4), 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"
}
] |
12.53°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(8, 6), B=(-1, 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"
}
] |
1.136
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-7, 5), B=(-5, -5), and C=(9, -7), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
109.44°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(0, 10), B=(5, -1), 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"
}
] |
114.44°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(8, 1), B=(-9, 7), and C=(3, -9), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
33.69°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-1, 6), B=(2, -5), and C=(5, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
105.26°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-5, -1), B=(-6, 2), 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"
}
] |
1.140
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(6, 6), B=(-3, 0), 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"
}
] |
22.38°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(8, -1), B=(0, 2), and C=(3, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
52.74°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-6, 2), B=(1, 1), and C=(6, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(10.500, 29.500)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(10, 5), B=(7, 3), 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"
}
] |
(-2.857, 6.286)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(6, -2), B=(-8, 6), 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"
}
] |
20.28°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-4, 10), B=(4, -9), and C=(-5, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-8.065, -8.290)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(0, -4), B=(-7, -3), and C=(4, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(1.895, -5.737)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(8, -9), B=(-4, 7), and C=(-5, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.531
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-1, 4), B=(5, -5), and C=(7, -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"
}
] |
0.985
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-1, -3), B=(-10, -6), and C=(1, -7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-0.857, -1.429)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-3, 7), B=(3, -8), and C=(-4, 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"
}
] |
10.67°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(3, -9), B=(-2, 3), and C=(5, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(9.889, -0.963)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(10, -9), B=(-2, 5), and C=(5, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
12.30°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(5, 4), B=(6, -9), and C=(9, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-19.241, 7.828)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(10, -4), B=(-5, 0), and C=(-5, -2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.919
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(4, 2), B=(-9, 0), and C=(-5, 6), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
47.56°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-6, 1), B=(5, 7), and C=(9, 10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.257
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(4, -2), B=(3, -2), and C=(8, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.450
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(1, 10), B=(-8, -3), and C=(5, -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"
}
] |
3.856
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-10, -9), B=(7, 2), and C=(-6, 8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
57.68°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(1, 1), B=(10, -3), and C=(1, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.794
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-7, 3), B=(-1, -4), and C=(-8, 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"
}
] |
4.40°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-6, 10), B=(-5, -8), and C=(9, 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"
}
] |
63.43°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-3, 0), B=(-6, -8), and C=(9, -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"
}
] |
77.04°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(7, -4), B=(6, 5), and C=(-4, -6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(7.762, -4.693)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-10, -10), B=(10, -5), and C=(-5, -8), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.73°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(8, 1), B=(5, -3), and C=(-4, 8). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.084
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(10, 10), B=(2, 5), and C=(-8, 8). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.948
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-7, -1), B=(-2, 3), and C=(-9, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-3.222, 2.778)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(1, -4), B=(-6, 2), and C=(-5, -1), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-11.000, -8.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(1, -6), B=(8, 5), and C=(-2, 9), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(5.174, 4.435)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-5, 8), B=(-1, -4), and C=(3, -2). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.871
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(7, 0), B=(3, -5), and C=(1, 2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(4.421, -0.737)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(1, 2), B=(5, 7), and C=(-3, -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"
}
] |
0.147
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-7, -8), B=(-5, 1), and C=(4, -5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-3.839, -3.258)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-5, -7), B=(1, -5), and C=(-2, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.464
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-6, -6), B=(-1, 8), and C=(5, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(0.333, 6.667)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(9, -9), B=(-1, -3), and C=(-1, 10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.818
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-2, -5), B=(10, 7), and C=(-7, -7), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(30.667, -44.667)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(0, 7), B=(-6, 4), and C=(-4, 3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
53.13°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-5, -4), B=(8, -7), and C=(9, -4), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(8.000, -8.333)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(9, -8), B=(9, -7), and C=(5, 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"
}
] |
162.90°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(5, -3), B=(3, -2), and C=(-5, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(1.143, 7.286)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(3, -6), B=(-3, 4), and C=(8, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.362
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(9, -10), B=(3, -5), and C=(2, -3), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
156.37°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(4, -5), B=(-9, 5), and C=(2, 7), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(1.853, 6.809)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(3, -4), B=(-10, -4), 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"
}
] |
90.00°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, 0), B=(-2, -5), and C=(2, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.711
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-6, 8), B=(-7, -8), and C=(-5, 1), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
8.95°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(2, -6), B=(-4, 6), 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"
}
] |
1.488
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(9, 8), B=(3, 5), and C=(-1, -6), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-11.778, 15.556)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-10, 6), B=(0, 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"
}
] |
(-7.000, 4.091)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(9, 4), B=(7, -10), and C=(4, 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"
}
] |
1.944
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(9, 5), B=(-2, 5), and C=(-4, -3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
2.550
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(2, 10), B=(-2, -1), and C=(-1, -9), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-39.116, 4.860)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(6, 10), B=(2, 6), 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"
}
] |
142.13°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(5, 5), B=(5, -5), 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"
}
] |
116.57°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-3, -9), B=(6, 7), and C=(10, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.996
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(1, 3), B=(-7, 0), and C=(7, 6), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(58.000, -130.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-6, 9), B=(0, -8), and C=(-5, 8), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(27.455, 19.455)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-6, -5), B=(4, 2), and C=(-8, -1). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.859
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 2), B=(-7, 3), and C=(5, 3). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.424
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-8, -7), B=(2, 6), and C=(9, 6), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
127.57°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(9, 10), B=(2, -6), 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"
}
] |
23.63°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(4, -5), B=(7, -3), and C=(-10, -2), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(3.027, -21.541)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(9, 1), B=(-3, 3), 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"
}
] |
2.013
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(4, -4), B=(-10, 7), 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"
}
] |
7.19°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-2, 1), B=(9, -9), and C=(10, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.015
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(0, -4), B=(5, -7), 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"
}
] |
2.196
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(8, -6), B=(-2, 4), and C=(8, -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"
}
] |
(13.000, 4.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-2, -4), B=(-4, -4), and C=(6, -10), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
30.96°
|
Puzzle
|
advanced_geometry
|
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