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Spherical Triangle: Medians and Centroid

Spherical Triangle: Medians and Centroid
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2014-07-23 08:10
TriangleCoordinate GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example illustrates the medians and centroid of a spherical triangle. Using GeoGebra commands such as Sphere, PointIn, Plane, IntersectPath, etc., the geometric properties of spherical triangles can be visually understood. Suitable for high school and above students and teachers.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → B
  4. PointIn(a) → C
  5. Plane(A, O, B) → b
  6. Plane(A, O, C) → c
  7. Plane(C, O, B) → d
  8. IntersectPath(b, a) → e
  9. IntersectPath(d, a) → f
  10. IntersectPath(c, a) → g
  11. Midpoint(A, B) → D
  12. Midpoint(B, C) → E
  13. Midpoint(C, A) → F
  14. OrthogonalLine(O, g) → i
  15. OrthogonalLine(O, e) → j
  16. OrthogonalLine(O, f) → h
  17. Plane(E, h) → k
  18. Plane(D, j) → l
  19. Plane(F, i) → m
  20. IntersectPath(l, a) → p
  21. IntersectPath(m, a) → q
  22. IntersectPath(k, a) → r
  23. Intersect(f, r, 2) → G
  24. Intersect(g, q, 2) → H
  25. Intersect(p, e, 1) → I
  26. Plane(A, O, G) → n
  27. OrthogonalLine(O, n) → s
  28. Plane(G, s) → o
  29. OrthogonalLine(O, o) → t
  30. Intersect(t, a, 2) → J
  31. CurveCartesian((x(G) * cos(t)) + (x(J) * sin(t)), (y(G) * cos(t)) + (y(J) * sin(t)), (z(G) * cos(t)) + (z(J) * sin(t)), t, 0, Angle[G, O, A]) → u
  32. Plane(O, I, C) → v
  33. OrthogonalLine(O, v) → a_1
  34. Plane(I, a_1) → w
  35. OrthogonalLine(O, w) → b_1
  36. Intersect(b_1, a, 2) → K
  37. CurveCartesian((x(I) * cos(t)) + (x(K) * sin(t)), (y(I) * cos(t)) + (y(K) * sin(t)), (z(I) * cos(t)) + (z(K) * sin(t)), t, 0, Angle[I, O, C]) → c_1
  38. Plane(B, O, H) → d_1
  39. OrthogonalLine(O, d_1) → e_1
  40. Plane(H, e_1) → f_1
  41. OrthogonalLine(O, f_1) → g_1
  42. Intersect(g_1, a, 2) → L
  43. CurveCartesian((x(H) * cos(t)) + (x(L) * sin(t)), (y(H) * cos(t)) + (y(L) * sin(t)), (z(H) * cos(t)) + (z(L) * sin(t)), t, 0, Angle[B, O, H]) → h_1

AI Drawing Prompt Reference

Draw a spherical triangle ABC, then draw its three medians AD, BE, and CF, which connect each vertex to the midpoint of the opposite side. Next, draw the intersection point G of these medians, which is the centroid of the spherical triangle.
The prompt above was compiled from the original drawing commands and is for reference only.