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Ellipse: Magic Sum by Tim Brzezinski

Ellipse: Magic Sum by Tim Brzezinski
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2016-02-06 09:26
Coordinate GeometryGeometric Construction
This example demonstrates an interesting property of ellipses: the sum of the distances from any point on the ellipse to its two foci is constant. Key mathematical concepts include the definition of an ellipse, foci, and vertices. The construction uses GeoGebra commands such as Focus, Vertex, Segment, Translate, etc. Suitable for high school mathematics education and learning about ellipse properties.

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