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Spherical Nine Point Circle

Spherical Nine Point Circle
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2014-07-03 09:33
CircleSolid GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example demonstrates the construction and properties of the Nine Point Circle on a sphere, involving spherical geometry and triangle geometry. Key GeoGebra commands and objects used include Sphere, PointIn, Mirror, Circle, Midpoint, Plane, OrthogonalLine, IntersectPath, and Intersect. Suitable for high school and above students to understand special circles in spherical geometry and their construction methods.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → B
  4. PointIn(a) → C
  5. Mirror(C, O) → C'
  6. Mirror(B, O) → B'
  7. Mirror(a, O) → a'
  8. Circle(C, A, C') → c
  9. Circle(B, A, B') → d
  10. Circle(B, C, B') → e
  11. Midpoint(B, A) → D
  12. Midpoint(A, C) → E
  13. Midpoint(C, B) → F
  14. Plane(B, O, C) → b
  15. Plane(C, O, A) → f
  16. Plane(A, O, B) → g
  17. OrthogonalLine(O, g) → h
  18. OrthogonalLine(O, f) → i
  19. OrthogonalLine(O, b) → j
  20. Plane(D, h) → k
  21. Plane(F, j) → l
  22. Plane(E, i) → m
  23. IntersectPath(k, a) → p
  24. IntersectPath(l, a) → q
  25. IntersectPath(m, a) → r
  26. Intersect(d, p, 2) → G
  27. Intersect(e, q, 1) → H
  28. Intersect(r, c, 2) → I
  29. Circle(G, H, I) → s

AI Drawing Prompt Reference

On a sphere, randomly select three points and reflect each across the sphere's center to get three more points. Construct two circles using each triangle's vertices and their corresponding reflected points. Find the midpoints of the triangle's sides, build three planes using these midpoints and the sphere's center, and draw the intersections of these planes with the sphere. Finally, use the intersections and circles' intersection points to draw a spherical nine-point circle.
The prompt above was compiled from the original drawing commands and is for reference only.