messages
listlengths 1
1
| ground_truth
stringlengths 5
21
| dataset
stringclasses 1
value | type
stringclasses 1
value |
|---|---|---|---|
[
{
"content": "In triangle ABC with coordinates A=(-8, -7), B=(2, -2), and C=(8, -5), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
126.87°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(9, 4), B=(6, -5), and C=(6, 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"
}
] |
1.257
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-9, -6), B=(-7, -4), and C=(10, -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"
}
] |
115.56°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-10, 1), B=(-4, 4), and C=(1, 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"
}
] |
114.78°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-1, 8), B=(2, -6), and C=(6, 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"
}
] |
2.715
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(7, 3), B=(-2, -9), and C=(-3, 9). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.894
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(1, 1), B=(0, -9), and C=(-1, -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"
}
] |
17.02°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-7, -1), B=(-8, 6), and C=(-2, 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"
}
] |
(-5.784, 0.459)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(2, 9), B=(-4, -8), and C=(6, -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"
}
] |
3.784
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(5, 3), B=(1, 9), 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"
}
] |
85.03°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-4, -8), B=(-7, -3), and C=(4, -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"
}
] |
(-7.000, -14.600)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(3, 8), B=(9, -5), and C=(3, 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"
}
] |
0.209
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-2, -6), B=(-8, 5), 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"
}
] |
91.13°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-1, -6), B=(-4, -2), and C=(-7, -6). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.500
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-3, -7), B=(5, -1), and C=(-4, 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"
}
] |
87.58°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-1, 0), B=(-1, 1), and C=(1, 4), find the measure (in degrees) of angle ABC. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
146.31°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-4, 3), B=(6, 4), and C=(-9, -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"
}
] |
(-11.127, 18.273)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(8, -4), B=(6, -2), and C=(-4, 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"
}
] |
(12.000, 16.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(6, 8), B=(4, -2), and C=(10, 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"
}
] |
2.112
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(9, 8), B=(-9, -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"
}
] |
(-73.000, 90.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-5, -3), B=(-3, -3), 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"
}
] |
26.57°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(1, 7), B=(2, -3), and C=(-6, -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"
}
] |
(5.446, -4.855)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(5, 10), B=(4, 8), and C=(-9, 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"
}
] |
0.660
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(10, -4), B=(4, -7), 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"
}
] |
51.34°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(6, 4), B=(-3, 1), and C=(9, -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"
}
] |
(5.788, 3.636)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(8, 8), B=(5, -8), and C=(-5, -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"
}
] |
74.05°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-3, 5), B=(-6, 3), and C=(-9, 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"
}
] |
164.74°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-5, -6), B=(3, 1), and C=(-7, -3), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-5.526, -4.684)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-10, -6), B=(7, 5), and C=(0, -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"
}
] |
(17.525, -30.085)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-7, -3), B=(-10, -6), and C=(10, 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"
}
] |
0.991
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(6, -6), B=(-9, 10), and C=(3, -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"
}
] |
0.068
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-2, 3), B=(7, 8), 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"
}
] |
(7.935, 0.516)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-9, -8), B=(3, 0), 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"
}
] |
(3.000, -6.500)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-4, -8), B=(10, -1), and C=(-10, 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"
}
] |
3.829
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(10, 3), B=(8, 6), and C=(-9, -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"
}
] |
(7.254, 8.836)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-1, 9), B=(-8, -10), and C=(0, 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"
}
] |
(21.815, -4.037)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(5, -2), B=(0, 1), and C=(10, 5), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(5.320, -2.800)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-7, -9), B=(2, -2), 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"
}
] |
(1.405, -7.949)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-10, -7), B=(2, 5), 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"
}
] |
54.46°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(2, -3), B=(9, -3), 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"
}
] |
2.467
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-8, 8), B=(-2, 10), and C=(-2, -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"
}
] |
71.57°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-3, 7), B=(1, 7), and C=(3, -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"
}
] |
1.498
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(5, -10), B=(-3, 2), and C=(4, 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"
}
] |
2.142
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(10, 5), B=(7, -5), and C=(-4, -3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(7.897, -6.569)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-8, -9), B=(2, -1), and C=(-9, 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"
}
] |
48.96°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(1, 10), B=(-6, 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"
}
] |
26.57°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-3, -2), B=(-3, -4), and C=(0, -5). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.638
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(8, -6), B=(4, -9), and C=(-2, -10). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
0.641
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(5, -4), B=(-8, -10), and C=(-8, -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"
}
] |
0.445
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(6, -7), B=(1, 9), 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"
}
] |
0.297
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(7, 10), B=(5, -2), 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"
}
] |
155.77°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-6, 9), B=(5, 10), 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"
}
] |
3.426
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(1, 3), B=(4, -7), 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"
}
] |
77.70°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(9, -9), B=(1, 7), and C=(-4, 5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(2.167, 8.083)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(3, -9), B=(2, 3), 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"
}
] |
21.80°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(10, -6), B=(-8, 3), 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"
}
] |
0.223
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-8, 2), 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"
}
] |
(-8.000, 2.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-4, -10), B=(-4, 4), and C=(-9, -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"
}
] |
(-13.000, -5.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(0, 7), B=(-8, 0), 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"
}
] |
2.212
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(3, -9), B=(8, 2), 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"
}
] |
1.328
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(6, 0), B=(-5, -8), and C=(2, 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"
}
] |
28.96°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -3), B=(8, -5), 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"
}
] |
0.996
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-7, -3), B=(-3, -8), and C=(1, 3), determine the orthocenter (intersection of altitudes). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-6.656, -3.125)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(9, -8), B=(-6, 0), 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"
}
] |
(17.556, 17.667)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(7, 1), B=(-6, -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"
}
] |
(-22.333, 15.667)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-4, -2), B=(3, -4), 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"
}
] |
88.09°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(8, -3), B=(8, 4), and C=(4, 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"
}
] |
1.313
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-2, 2), B=(-4, -2), and C=(3, -10). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
1.566
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(-1, -8), B=(-7, 10), and C=(1, -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"
}
] |
(-20.000, -16.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, -10), B=(-9, -2), and C=(9, 2). For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
3.128
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-7, -6), B=(-4, 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"
}
] |
30.96°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(-1, 9), B=(1, 0), 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"
}
] |
(46.000, 9.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(6, 7), B=(-6, 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"
}
] |
119.45°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(3, 1), B=(3, 9), 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"
}
] |
2.775
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(0, 6), B=(-3, 2), 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"
}
] |
(-24.000, 9.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-9, 5), B=(-3, 9), and C=(8, -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"
}
] |
2.861
|
Puzzle
|
advanced_geometry
|
[
{
"content": "For triangle with vertices A=(5, -6), B=(-3, -3), and C=(5, 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"
}
] |
(0.500, -3.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(2, -5), B=(-1, -7), 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"
}
] |
0.618
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-4, -1), B=(-7, -7), 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"
}
] |
1.146
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(3, 7), B=(2, -6), and C=(-10, -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"
}
] |
3.741
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-2, 6), B=(-5, -3), 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"
}
] |
40.24°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(7, -8), B=(3, 9), and C=(5, 10), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(-1.263, 8.526)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "In triangle ABC with coordinates A=(-7, 4), B=(10, -3), and C=(-4, -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"
}
] |
18.29°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-10, 7), B=(6, 2), and C=(-2, 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"
}
] |
1.608
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(7, 9), B=(-8, 8), and C=(1, 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"
}
] |
22.25°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(-2, 8), B=(-1, -5), and C=(3, 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"
}
] |
34.14°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(2, -5), B=(9, -1), and C=(10, 4), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(16.839, -7.968)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Find the incircle radius of triangle ABC whose vertices are A=(-8, 0), B=(-8, -8), and C=(3, 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"
}
] |
2.517
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(7, 7), B=(3, 5), and C=(-4, -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"
}
] |
143.13°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-6, -4), B=(-9, 6), and C=(3, 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"
}
] |
3.112
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-9, 10), B=(3, 9), and C=(1, -7). Compute the radius of its incircle. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
4.051
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(5, -6), B=(10, -5), and C=(-7, -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"
}
] |
(22.429, -154.143)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(6, -1), B=(6, 8), and C=(-5, 4), determine the angle at B in degrees. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
70.02°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given a triangle with vertices A=(0, -5), B=(4, 10), and C=(-6, 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"
}
] |
58.37°
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-9, -7), B=(5, 8), 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"
}
] |
3.961
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(-7, -3), B=(-2, -4), 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"
}
] |
1.989
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(5, 6), B=(2, 6), 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"
}
] |
0.769
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Consider triangle ABC with coordinates A=(7, 2), B=(8, -8), 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"
}
] |
0.099
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(6, 2), B=(1, -8), and C=(4, -5), find the coordinates of its orthocenter. For all geometry problems:\n1. Give coordinates in the form (x, y)\n2. Round decimal answers to 3 decimal places\n3. Use the degree symbol ° for angles\n4. Return only the angle, coordinates, or radius as your answer.\n",
"role": "user"
}
] |
(22.000, -14.000)
|
Puzzle
|
advanced_geometry
|
[
{
"content": "Given triangle ABC with coordinates A=(8, -9), B=(-5, 7), and C=(-1, -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"
}
] |
(-41.000, -33.500)
|
Puzzle
|
advanced_geometry
|
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