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Ellipse: Magic Sum

Ellipse: Magic Sum
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2020-03-27 22:11
VectorCoordinate GeometryParametric Equation
This example demonstrates the geometric properties of an ellipse, particularly the relationship between the distances from the foci, vertices, and internal points. Core mathematical concepts include the definition of an ellipse, focal properties, vector operations, and parametric equations. Construction highlights involve GeoGebra commands/objects like Focus, Vertex, Segment, Translate, Point, Vector, Dilate, Angle, Rotate, Circle, Center, and Intersect. Suitable for high school and above students to explore elliptical geometry, and for teaching parametric equations and vector operations.

GeoGebra Commands & Parameters

  1. Focus(c) → F_1, F_2
  2. Vertex(c) → C, D, E, F
  3. Segment(C, D) → d
  4. Translate(C, (0, b + 4)) → C'
  5. Translate(D, (0, b + 4)) → D'
  6. Point(c) → G
  7. Segment(G, F_1) → e
  8. Segment(G, F_2) → f
  9. Translate(C, (b + 4, 0)) → C'_1
  10. Vector(C, C') → u
  11. Vector(D, D') → v
  12. Dilate(C, If[0 < n < 1, 1 - n, 0], C') → C'_2
  13. Dilate(D, If[0 < n < 1, 1 - n, 0], D') → D'_2
  14. Dilate(F_1, If[1 < n < 2, 2 - n, 1], C') → A'
  15. Vector(F_1, A') → w
  16. Translate(G, w) → G'
  17. Segment(A', G') → m
  18. Angle(F_2, F_1, G) → α
  19. Rotate(A', If[1 < n < 2, ((-(2 - n)) * α)], A') → A''
  20. Rotate(A', If[1 < n < 2, ((-(n - 2)) * α), 0], A'') → A''_1
  21. Rotate(A', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → A''_2
  22. Rotate(G', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → G''
  23. Circle(C', e) → q
  24. Segment(F_1, G) → s
  25. Dilate(F_2, If[2 < n < 3, 3 - n, 1], D') → B'
  26. Vector(F_2, B') → t
  27. Translate(G, t) → G'_2
  28. Translate(F_2, t) → B'_1
  29. Segment(G'_2, B'_1) → f'
  30. Angle(G, F_2, F_1) → β
  31. Rotate(G'_2, If[2 < n < 3, ((n - 2) * β)], B') → L
  32. Rotate(B'_1, If[2 < n < 3, ((n - 2) * β)], B') → M
  33. Segment(G, F_2) → b_1
  34. Center(c) → A
  35. Segment(A, G) → c_1
  36. Segment(A, G) → d_1
  37. Dilate(A, If[1 < n < 2, 2 - n, 0], C') → A'_1
  38. Vector(F_1, A'_1) → u_1
  39. Translate(A, u_1) → A'_2
  40. Angle(D, F_1, G) → γ
  41. Rotate(A'_2, ((-(n - 1)) * γ), A'_2) → B
  42. Circle(C, e) → p
  43. Intersect(p, d, 1) → H
  44. Dilate(F_1, If[0 < n < 1, 1 - n, 1], C) → I
  45. Vector(F_1, I) → l
  46. Translate(G, l) → G'_1
  47. Translate(F_1, l) → J
  48. Segment(G'_1, J) → e'
  49. Rotate(G'_1, If[0 < n < 1, (-((n * α))), (-α)], I) → K
  50. Rotate(J, If[0 < n < 1, (-((n * α))), (-α)], I) → J'
  51. Segment(K, J') → e''
  52. Segment(C, H) → r
  53. Dilate(F_2, If[1 < n < 2, 2 - n, 1], D) → N
  54. Vector(F_2, N) → v_1
  55. Translate(G, v_1) → G'_3
  56. Translate(F_2, v_1) → O
  57. Segment(G'_3, O) → f'_1
  58. Rotate(G'_3, If[1 < n < 2, ((n - 1) * β), β], N) → P
  59. Rotate(O, If[1 < n < 2, ((n - 1) * β), β], N) → O'
  60. Segment(P, O') → a_1
  61. Segment(D, H) → e_1
  62. Dilate(F_1, 1 - n, C) → Q
  63. Vector(F_1, Q) → h
  64. Translate(A, h) → A'_3
  65. Translate(G, h) → G'_4
  66. Segment(A'_3, G'_4) → i
  67. Angle(D, Center[c], G) → δ
  68. Rotate(A'_3, If[0 < n < 1, (-((n * δ)))], Q) → R
  69. Rotate(G'_4, If[0 < n < 1, (-((n * δ)))], Q) → S
  70. Segment(R, S) → i'
  71. Dilate(F_2, If[1 < n < 2, 2 - n], D) → T
  72. Vector(F_1, T) → w_1
  73. Translate(A, w_1) → A'_4
  74. Translate(G, w_1) → G'_5
  75. Segment(A'_4, G'_5) → f_1
  76. Angle(G, Center[c], C) → ε
  77. Rotate(A'_4, If[1 < n < 2, ((n - 1) * ε)], A'_4) → U
  78. Rotate(G'_5, If[1 < n < 2, ((n - 1) * ε)], A'_4) → V
  79. Segment(U, V) → g_1