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Angle Sum of Triangle on a Sphere

Angle Sum of Triangle on a Sphere
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2018-12-09 20:41
TrigonometrySolid GeometryGeometric Construction
This example demonstrates the angle sum property of a spherical triangle. It involves core mathematical concepts like spherical geometry, angle measurement, and non-Euclidean geometry. Construction highlights include using GeoGebra commands such as Intersect, Sphere, IntersectPath, Point, CircleArc, Midpoint, Line, etc., to draw and calculate the angle sum. Suitable for high school students and geometry enthusiasts studying spherical geometry properties.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xAxis) → A
  2. Sphere(A, 1) → a
  3. Intersect(xAxis, a, 2) → B
  4. IntersectPath(xOyPlane, a) → c
  5. Point(c) → C
  6. PointIn(a) → D
  7. CircleArc(A, B, D) → d
  8. CircleArc(A, C, D) → e
  9. Intersect(xAxis, d, 1) → E
  10. CircleArc(A, E, C) → f
  11. Midpoint(D, E) → F
  12. Line(A, F) → g
  13. If(0 ≤ n < 1, Rotate[D, (n * π), g], B) → G
  14. If(0 ≤ n < 1, Rotate[B, (n * π), g], D) → H
  15. If(0 ≤ n < 1, Rotate[C, (n * π), g], Rotate[C, π, g]) → I
  16. CircleArc(A, G, H) → h
  17. CircleArc(A, G, I) → k
  18. CircleArc(A, H, I) → p
  19. Midpoint(D, C) → J
  20. Midpoint(I, E) → K
  21. Plane(K, J, A) → b
  22. OrthogonalLine(A, b) → i
  23. If(2 ≤ n < 3, Rotate[e, ((-(n - 2)) * Angle[K, A, J]), i], k) → q
  24. If(2 ≤ n < 3, Rotate[D, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[D, (-Angle[J, A, K]), i]) → L
  25. If(2 ≤ n < 3, Rotate[C, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[C, (-Angle[J, A, K]), i]) → M
  26. If(2 ≤ n < 3, Rotate[B, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[B, (-Angle[J, A, K]), i]) → N
  27. CircleArc(A, L, N) → r
  28. CircleArc(A, M, N) → s
  29. Line(A, D) → j
  30. Line(A, C) → l
  31. OrthogonalPlane(D, j) → m
  32. Plane(D, A, E) → o
  33. Plane(D, A, C) → t
  34. IntersectPath(t, m) → f_1
  35. IntersectPath(o, m) → g_1
  36. Circle(D, 0.2, j) → c_1
  37. Intersect(g_1, c_1) → O, P
  38. Intersect(f_1, c_1) → Q, R
  39. CircleSector(D, P, R) → d_1
  40. If(0 ≤ n < 1, Rotate[d_1, (n * π), g], Rotate[d_1, π, g]) → e_1
  41. If(2 ≤ n < 3, Rotate[d_1, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[d_1, (-Angle[J, A, K]), i]) → h_1
  42. OrthogonalPlane(E, xAxis) → u
  43. IntersectPath(o, u) → i_1
  44. IntersectPath(xOyPlane, u) → j_1
  45. Circle(E, 0.2, xAxis) → k_1
  46. Intersect(i_1, k_1) → S, T
  47. Intersect(j_1, k_1) → U, V
  48. CircleSector(E, S, U) → p_1
  49. If(0 ≤ n < 1, Rotate[p_1, (n * π), g], Rotate[p_1, π, g]) → q_1
  50. If(2 ≤ n < 3, Rotate[p_1, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[p_1, (-Angle[J, A, K]), i]) → r_1
  51. Plane(A, D, C) → v
  52. OrthogonalPlane(C, l) → w
  53. IntersectPath(v, w) → l_1
  54. IntersectPath(w, xOyPlane) → m_1
  55. Circle(C, 0.2, l) → s_1
  56. Intersect(l_1, s_1) → W, Z
  57. Intersect(m_1, s_1) → A_1, B_1
  58. CircleSector(C, Z, A_1) → t_1
  59. If(0 ≤ n < 1, Rotate[t_1, (n * π), g], Rotate[t_1, π, g]) → c_2
  60. If(2 ≤ n < 3, Rotate[t_1, ((-(n - 2)) * Angle[J, A, K]), i], Rotate[t_1, (-Angle[J, A, K]), i]) → d_2

AI Drawing Prompt Reference

Draw a spherical triangle on a sphere to demonstrate the property that the sum of its interior angles exceeds 180 degrees. Start by sketching the equator and a meridian to define the triangle's vertices on the sphere's surface. Connect these points to form the spherical triangle, then draw arcs for each interior angle. Use rotations and auxiliary lines to indicate the measurement of the angle sum, highlighting it with differently colored arcs to visually show the difference between spherical and Euclidean geometry.
The prompt above was compiled from the original drawing commands and is for reference only.