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Dandelin's Theorem in Stereometry

Dandelin's Theorem in Stereometry
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2015-07-14 23:25
Coordinate GeometrySolid GeometryGeometric Construction
This example illustrates Dandelin's theorem, showing the relationship between the inscribed sphere and the conic section in a cone. Key mathematical concepts include cones, conic sections, and solid geometry. Using GeoGebra commands such as ConeInfinite, Rotate, and IntersectPath, the geometric meaning of the theorem is intuitively understood. Suitable for high school solid geometry courses.

GeoGebra Commands & Parameters

  1. ConeInfinite((0, 0, 0), zAxis, α) → a
  2. Axes(a) → d, e, f
  3. Rotate(f, (-α), yAxis) → f'_1
  4. Point(f'_1) → A
  5. Rotate(f, α, yAxis) → f'
  6. Point(f') → B
  7. Plane(A, B_1, B) → b
  8. IntersectPath(b, a) → c
  9. Axes(c) → i, j
  10. AngularBisector(f', i) → b2, b1
  11. Center(a) → O
  12. Vector(O, B) → vOB
  13. Angle(Vector[A, B], Vector[A, O]) → β
  14. Vector(O, A) → vOA
  15. Rotate(vOA, β / 2, A, k) → vOA'
  16. Angle(Vector[B, A], Vector[O, B]) → α1
  17. Rotate(vOB, α1 / 2, B, k) → vOB'
  18. Ray(B, B') → g
  19. Intersect(f, b1) → G
  20. OrthogonalLine(G, i, k) → h
  21. Intersect(i, h) → H
  22. Circle(G, H, k) → p
  23. Sphere(G, H) → l
  24. Ray(A, A') → m
  25. Intersect(f, b2) → L
  26. OrthogonalLine(L, i, k) → n
  27. Intersect(i, n) → M
  28. Circle(L, M, k) → s
  29. Point(c) → P
  30. Line(O, P) → a_1
  31. Segment(M, P) → b_1
  32. Segment(P, H) → e_1
  33. Sphere(L, M) → l'
  34. OrthogonalLine(L, f'_1, k) → q
  35. OrthogonalLine(G, f'_1, k) → r
  36. Intersect(f'_1, r) → C
  37. Intersect(q, f'_1) → D
  38. Circle(f, C) → t
  39. Circle(f, D) → c_1
  40. Line(O, P) → d_1
  41. Intersect(d_1, c_1) → E,
  42. Intersect(d_1, t) → F,
  43. Segment(P, F) → f_1
  44. Segment(E, P) → g_1
  45. Segment(O, E) → h_1