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Dandelin's Theorem Demonstration

Dandelin's Theorem Demonstration
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2015-03-14 00:35
CircleAngleSolid GeometryGeometric Construction
This example dynamically illustrates Dandelin's theorem, revealing the relationship between circles and conic sections in a cone; core math concepts include cones, angles, quadratic equations, and solid geometry; construction highlights involve commands/objects like ConeInfinite, Center, IntersectPath, Point, Axes, Plane, Vector, Angle, Rotate, Ray, Intersect, OrthogonalLine, Circle, Sphere, Line, Segment from GeoGebra; suitable for high school and above students studying solid geometry and conic sections.

GeoGebra Commands & Parameters

  1. ConeInfinite((0, 0, 0), zAxis, 30*°) → a
  2. Center(a) → O
  3. IntersectPath(k, a) → q
  4. Point(q) → K
  5. Axes(a) → d, e, f
  6. Point(Line[O, K]) → A
  7. Point(q) → F
  8. Point(Line[O, F]) → B
  9. Plane(A, B_1, B) → b
  10. IntersectPath(b, a) → c
  11. Axes(c) → i, j
  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]) → α
  17. Rotate(vOB, α / 2, B, k) → vOB'
  18. Ray(B, B') → g
  19. Intersect(f, g) → G
  20. OrthogonalLine(G, i, k) → h
  21. Intersect(i, h) → H
  22. Circle(G, H, k) → p
  23. Sphere(G, H) → l
  24. Intersect
  25. Intersect(q, p, 3) → I
  26. Circle(f, I) → r
  27. Ray(A, A') → m
  28. Intersect(f, m) → L
  29. OrthogonalLine(L, i, k) → n
  30. Intersect(i, n) → M
  31. Circle(L, M, k) → s
  32. Intersect
  33. Intersect(q, s, 1) → N
  34. Sphere(L, N) → o
  35. Point(c) → P
  36. Circle(f, N) → t
  37. Line(O, P) → a_1
  38. Intersect(a_1, r, 1) → Q
  39. Intersect(a_1, t, 1) → R
  40. Segment(M, P) → b_1
  41. Segment(P, R) → c_1
  42. Segment(Q, P) → d_1
  43. Segment(P, H) → e_1
  44. Plane(M, R, P) → u
  45. IntersectPath(u, o) → f_1
  46. Plane(H, P, Q) → v
  47. IntersectPath(v, l) → g_1

AI Drawing Prompt Reference

Demonstrate Dandelin's theorem on an infinite cone. Start by drawing the cone's axis and a circle on its surface. From the cone's vertex, draw two tangents to the circle, connecting them to the circle's points of tangency. Form two intersecting planes using these points and the vertex. On these planes, draw circles intersecting the cone, marking their centers and intersection points. Finally, draw the lines of intersection between these circles and the cone, along with relevant auxiliary lines and angles, to reveal the relationship between the cone, circles, and conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.