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Conics - Dandelin Proof

Conics - Dandelin Proof
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2014-11-07 01:49
Coordinate GeometrySolid GeometryGeometric Construction
In collections Selected Examples
This example demonstrates conic sections formed by the intersection of a plane and a cone using Dandelin's method. Key concepts include ellipses, hyperbolas, and parabolas. GeoGebra commands such as PointIn, Mirror, Angle, and Line are used for geometric construction. Suitable for high school mathematics education to help students understand the geometric properties of conic sections.

GeoGebra Commands & Parameters

  1. PointIn(xz) → K
  2. Mirror(K, zAxis) → K'
  3. Angle(K, O, K') → α
  4. Line((-3, 0, 0), (-3, 0, 1) + K) → h_1
  5. Point(h_1) → E_3
  6. Plane(E_1, E_2, E_3) → ε
  7. Line(K, O) → g_1
  8. Line(K', O) → g_2
  9. InfiniteCone(O, zAxis, α / 2) → a
  10. Midpoint(E_1, E_2) → B_1
  11. Line(E_3, B_1) → b
  12. Intersect(b, g_1) → A
  13. Intersect(b, g_2) → B
  14. Line(O, (0, 0, 1)) → zA
  15. Circle(A, 1, xz) → c
  16. Intersect(g_1, c) → C, D
  17. Intersect(b, c) → E, F
  18. Circle(E, 1, xz) → d
  19. Circle(D, 1, xz) → e
  20. Intersect(e, d) → G, H, ,
  21. Line(A, G) → w_1
  22. OrthogonalLine(A, w_1, xz) → w_2
  23. Intersect(zA, w_1) → M_1
  24. Intersect(zA, w_2) → M_2
  25. OrthogonalLine(M_1, g_1, xz) → f
  26. OrthogonalLine(M_2, g_1, xz) → g
  27. Intersect(g_1, f) → I
  28. Intersect(g_1, g) → J
  29. Segment(M_1, I) → r_1
  30. Segment(M_2, J) → r_2
  31. Sphere(M_1, I) → sp1
  32. Sphere(M_2, J) → sp2
  33. Circle(zA, J) → p
  34. Circle(zA, I) → q
  35. Point(p) → L
  36. Line(L, O) → j
  37. Intersect(j, q) → M,
  38. Intersect(j, ε) → P
  39. IntersectPath(ε, a) → r
  40. OrthogonalLine(M_1, b, xz) → l
  41. OrthogonalLine(M_2, b, xz) → m
  42. Intersect(b, l) → F_1
  43. Intersect(b, m) → F_2
  44. Segment(L, P) → s_1
  45. Segment(P, F_2) → s_2
  46. Intersect(j, q, 1) → N
  47. Segment(P, N) → t_1
  48. Segment(P, F_1) → t_2
  49. Circle(M_1, I, xz) → k
  50. Circle(M_2, J, xz) → s

AI Drawing Prompt Reference

First, draw conic sections formed by the intersection of a plane and a cone, such as ellipses, hyperbolas, or parabolas. Construct by first defining the cone and intersecting plane, finding intersection points, then drawing the conic sections. Later, add auxiliary lines like foci and directrices to illustrate geometric properties.
The prompt above was compiled from the original drawing commands and is for reference only.