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Triangle on a Sphere

Triangle on a Sphere
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2015-02-18 18:36
TriangleCoordinate GeometrySolid Geometry
This example demonstrates a triangle on a sphere. Key mathematical concepts include spherical geometry and basic properties of triangles. Using GeoGebra commands such as Intersect, PointIn, and IntersectConic, a triangle on a sphere can be constructed. Suitable for high school and above students or teachers for teaching or self-study of spherical geometry.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → O
  2. Sphere(O, R) → a
  3. PointIn(a) → Cfree
  4. PointIn(a) → B
  5. PointIn(a) → A
  6. Sphere(A, B) → d
  7. IntersectConic(a, d) → f
  8. Sphere(B, A) → b
  9. IntersectConic(a, b) → e
  10. Intersect
  11. Intersect(e, f, 1) → Cequalateral
  12. Plane(A, O, B) → g
  13. OrthogonalLine(O, g) → axisAB
  14. Circle(axisAB, B) → gab
  15. Tangent(B, gab) → tanBA2, tanBA1
  16. OrthogonalPlane(O, tanBA1) → h
  17. IntersectPath(h, a) → k
  18. Point(k) → Cright
  19. If(free, Cfree, If[right, Cright, Cequalateral]) → C
  20. Plane(O, A, C) → c
  21. OrthogonalLine(O, c) → axisAC
  22. Circle(axisAC, C) → gac
  23. Plane(O, B, C) → i
  24. OrthogonalLine(O, i) → axisBC
  25. Circle(axisBC, B) → gbc
  26. Tangent(B, gbc) → tanBC1, tanBC2
  27. Tangent(A, gab) → tanAB1, tanAB2
  28. Tangent(A, gac) → tanAC2, tanAC1
  29. UnitVector(tanBA2) → v1
  30. UnitVector(tanBC1) → v2
  31. Angle(B1, B, B2) → β
  32. UnitVector(tanAC2) → w1
  33. UnitVector(tanAB1) → w2
  34. Angle(A2, A, A1) → γ
  35. Point(gbc) → BC
  36. Point(gac) → AC
  37. Point(gab) → AB
  38. CircumcircleArc(B, AB, A) → ab
  39. Segment(O, A) → OA
  40. Segment(O, B) → OB
  41. Midpoint(A, B) → D
  42. CircumcircleArc(C, AC, A) → ac
  43. CircumcircleArc(B, BC, C) → bc
  44. Tangent(C, gac) → tanCA1, tanCA2
  45. Tangent(C, gbc) → tanCB1, tanCB2
  46. Angle(C2, C, C1) → α

AI Drawing Prompt Reference

Draw a triangle on a sphere to demonstrate spherical geometry. Start by identifying the sphere's center and the intersection of the equator with a vertical axis. Randomly select three points on the sphere to form the triangle's vertices. Then, draw circles centered at each vertex that intersect the sphere, creating three great circle arcs. Connect these intersection points to form the spherical triangle. Finally, label the triangle's edges and vertices, and display lines from the center of the sphere to each vertex.
The prompt above was compiled from the original drawing commands and is for reference only.