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Spherical Incenter and Incircle

Spherical Incenter and Incircle
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2014-07-03 03:19
CircleSolid GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example demonstrates the geometric construction of the spherical incenter and incircle in spherical geometry. Key concepts include spherical geometry, incircle, incenter, and geometric constructions. It utilizes commands like Sphere, PointIn, Plane, IntersectPath, Midpoint, Line, Intersect, Circle, OrthogonalPlane, and OrthogonalLine in GeoGebra. Suitable for high school and above students to learn about spherical geometry and properties of the incircle, it can be used for geometric construction teaching.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → B
  4. PointIn(a) → C
  5. Plane(O, A, B) → b
  6. Plane(B, O, C) → c
  7. Plane(O, C, A) → d
  8. IntersectPath(b, a) → e
  9. IntersectPath(c, a) → f
  10. IntersectPath(d, a) → g
  11. Midpoint(B, C) → D
  12. Midpoint(C, A) → E
  13. Midpoint(A, B) → F
  14. Line(O, A) → h
  15. Line(O, B) → i
  16. Line(O, C) → j
  17. Plane(D, h) → k
  18. Plane(F, j) → l
  19. Plane(E, i) → m
  20. IntersectPath(k, a) → p
  21. IntersectPath(m, a) → q
  22. IntersectPath(l, a) → r
  23. Intersect
  24. Intersect(f, p, 2) → G
  25. Intersect
  26. Intersect(q, g, 2) → H
  27. Intersect
  28. Intersect(e, r, 1) → I
  29. Line(O, C) → n
  30. Line(O, A) → s
  31. Circle(n, G) → t
  32. Circle(i, I) → c_1
  33. Circle(s, H) → d_1
  34. Intersect
  35. Intersect(d_1, e, 2) → J
  36. Midpoint(J, H) → K
  37. Intersect
  38. Intersect(c_1, f, 1) → L
  39. Intersect
  40. Intersect(g, t, 1) → M
  41. Midpoint(I, L) → N
  42. Midpoint(G, M) → P
  43. Plane(A, K, O) → o
  44. Plane(B, N, O) → u
  45. Plane(C, P, O) → v
  46. IntersectPath(o, a) → e_1
  47. IntersectPath(v, a) → f_1
  48. IntersectPath(u, a) → g_1
  49. Intersect
  50. Intersect(e_1, f_1) → Q, R, ,
  51. OrthogonalPlane(O, n) → w
  52. Plane(O, A, C) → a_1
  53. OrthogonalLine(O, a_1) → b_1
  54. Plane(Q, b_1) → h_1
  55. IntersectPath(h_1, a) → k_1
  56. Intersect(k_1, g, 2) → S
  57. Line(O, Q) → i_1
  58. Circle(i_1, S) → p_1

AI Drawing Prompt Reference

Select three random points on a sphere, construct planes through each pair of points and the sphere's center, and find their intersections with the sphere. Then, find the midpoints of each pair of intersection points, and construct new planes through the sphere's center and these midpoints. Find the intersections of these new planes with the sphere. Construct circles centered at the sphere's center and passing through these intersection points. Finally, find the intersections of these circles with the sphere and connect them to the sphere's center to obtain the inscribed circle and incenter.
The prompt above was compiled from the original drawing commands and is for reference only.