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Sine Theorem in Spherical Triangles

Sine Theorem in Spherical Triangles
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2017-12-17 05:26
TriangleTrigonometryGeometric Construction
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This example demonstrates the sine theorem in spherical triangles. The sine theorem is an important concept in spherical trigonometry, used to solve the relationships between sides and angles in spherical triangles. Using various GeoGebra commands such as Sphere, Circle, PointIn, CircleArc, Tangent, etc., it dynamically shows the relationships between the sides and angles in spherical triangles. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Sphere(O, r) → esf
  2. Circle(zAxis, E) → c_2
  3. Circle(yAxis, E) → d
  4. PointIn(esf) → B
  5. PointIn(esf) → C
  6. PointIn(esf) → A
  7. CircleArc(O, A, B) → c'
  8. CircleArc(O, C, B) → a'
  9. CircleArc(O, A, C) → b'
  10. Tangent(B, c') → h,
  11. Tangent(B, a') → i,
  12. Tangent(C, b') → m,
  13. Tangent(C, a') → n,
  14. Tangent(A, c') → s,
  15. Tangent(A, b') → t,
  16. Intersect(n, i) → T
  17. Intersect(s, h) → U
  18. Intersect(m, t) → V
  19. Ray(A, V) → j
  20. Ray(C, V) → k
  21. Ray(B, U) → l
  22. Ray(A, U) → p
  23. Ray(B, T) → q
  24. Ray(C, T) → f_1
  25. Angle(p, j) → α
  26. Angle(l, q) → β
  27. Angle(f_1, k) → γ
  28. Line(N, S) → g_1
  29. Circle(xAxis, N) → c_3
  30. Angle(C, O, A) → b
  31. Angle(B, O, C) → a
  32. Angle(A, O, B) → c
  33. CircleSector(O, C, A) → b_1
  34. CircleSector(O, B, C) → a_1
  35. CircleSector(O, A, B) → c_1

AI Drawing Prompt Reference

Draw a spherical triangle with three vertices A, B, and C and their corresponding spherical arcs c', a', and b'. Construct tangents to arcs c', a', and b' at points B, C, and A, respectively, and find their intersection points T, U, and V. Use these points and vertices to construct three rays j, k, and l, and three lines p, q, and f_1. Finally, construct three spherical arcs b_1, a_1, and c_1 with vertices A, B, C, and the center O, and three central angles b, a, and c.
The prompt above was compiled from the original drawing commands and is for reference only.