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Orthocenter and Circumcenter: Sphere vs Plane

Orthocenter and Circumcenter: Sphere vs Plane
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2018-03-25 23:10
TriangleCircleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the concepts of orthocenter and circumcenter in spheres and planes. The orthocenter is the point where the three altitudes of a triangle intersect, while the circumcenter is the point where the perpendicular bisectors of the triangle's sides intersect. Using GeoGebra commands such as Sphere, PointIn, Plane, IntersectPath, etc., this example provides a visual understanding of these geometric concepts. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Sphere(A, 1) → a
  2. PointIn(a) → B
  3. PointIn(a) → C
  4. PointIn(a) → D
  5. Plane(A, B, C) → b
  6. Plane(A, C, D) → c
  7. Plane(A, B, D) → d
  8. IntersectPath(b, a) → e
  9. IntersectPath(d, a) → f
  10. IntersectPath(c, a) → g
  11. OrthogonalLine(A, d) → h
  12. OrthogonalLine(A, b) → i
  13. OrthogonalLine(A, c) → j
  14. Plane(B, j) → k
  15. Plane(C, h) → l
  16. IntersectPath(l, a) → p
  17. IntersectPath(k, a) → q
  18. Plane(D, i) → m
  19. IntersectPath(m, a) → r
  20. Intersect(r, q, 1) → H
  21. Circle(B, C, D) → s
  22. Plane(B, C, D) → n
  23. OrthogonalLine(A, n) → t
  24. Intersect(t, a, 2) → O
  25. Plane(O, j) → o
  26. Plane(O, i) → u
  27. Plane(O, h) → v
  28. IntersectPath(u, a) → c_1
  29. IntersectPath(v, a) → d_1
  30. IntersectPath(o, a) → e_1
  31. Intersect(g, e_1, 2) → E
  32. Intersect(c_1, e, 1) → F
  33. Intersect(f, d_1, 2) → G
  34. Circle(F, G, E) → f_1
  35. Polygon(I, J, K) → t1, k_1, i_1, j_1
  36. OrthogonalLine(I, i_1) → g_1
  37. OrthogonalLine(J, j_1) → h_1
  38. LineBisector(J, K) → l_1
  39. LineBisector(K, I) → m_1
  40. Intersect(g_1, h_1) → L
  41. Intersect(l_1, m_1) → M