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Dynamic Geometric Problem Demonstration

Dynamic Geometric Problem Demonstration
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2018-04-26 10:43
VectorFunctionCoordinate GeometryTransformationGeometric Construction
This example dynamically demonstrates a geometric problem posted by Antonio Gutierrez, involving the intersection of parabolas and hyperbolas along with constructions. Key math concepts include coordinate geometry, geometric transformations, midpoint, rotation, dilation, and more. The construction utilizes commands and objects such as Ray, Polygon, Midpoint, Segment, Dilate, Rotate, Circle, Intersect, CircleSector, Vector, Translate, Line, and OrthogonalLine. Suitable for high school students and teachers for understanding complex geometric solutions.

GeoGebra Commands & Parameters

  1. Ray((0, 0), (1, 0)) → h_1
  2. Point(h_1) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  4. Midpoint(A, B) → F
  5. Segment(A, C) → j
  6. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B) → A'
  7. Segment(A, A') → k
  8. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A'_1
  9. Rotate(A', If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A''
  10. Segment(A'_1, A'') → k'
  11. Rotate(A'_1, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → J
  12. Rotate(A'', If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → A'''
  13. Segment(J, A''') → k''
  14. Rotate(J, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → J'
  15. Rotate(A''', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → K
  16. Segment(J', K) → k'''
  17. Circle(B, (0.15 * Distance[A, B])) → c
  18. Segment(A, B) → l
  19. Intersect(c, l) → L,
  20. Rotate(L, If[4 < a ≤ 5, ((-(a - 4)) * π / 2), ((-π)) / 2], B) → L'
  21. CircleSector(B, L', L) → d
  22. Dilate(B, If[6 < a ≤ 7, 7 - a, 0], C) → B'
  23. Vector(B, B') → u
  24. Translate(d, u) → d'
  25. Rotate(d', If[6 < a ≤ 7, ((a - 6) * π / 2), π / 2], B') → d''
  26. Dilate(C, If[7.5 < a ≤ 8.5, 8.5 - a, 0], D) → C'
  27. Vector(C, C') → v
  28. Translate(d'', v) → d'''
  29. Rotate(d''', If[7.5 < a ≤ 8.5, ((a - 7.5) * π / 2), π / 2], C') → e
  30. Dilate(D, If[9 < a ≤ 10, 10 - a, 0], A) → D'
  31. Vector(D, D') → w
  32. Translate(e, w) → e'
  33. Rotate(e', If[9 < a ≤ 10, ((a - 9) * π / 2), π / 2], D') → e''
  34. Dilate(A, If[10.5 < a ≤ 11.5, 11.5 - a, 0], B) → A'_2
  35. Vector(A, A'_2) → b
  36. Translate(e'', b) → e'''
  37. Rotate(e''', If[10.5 < a ≤ 11.5, ((a - 10.5) * π / 2), π / 2], A'_2) → p
  38. Line(N, B) → q
  39. OrthogonalLine(A, q) → r
  40. Dilate(B, If[11.5 < a ≤ 12.5, 12.5 - a, 0], G) → B'_1
  41. Segment(B, B'_1) → n
  42. Ray(G, O) → s
  43. Dilate(D, If[11.5 < a ≤ 12.5, 12.5 - a, 0], N) → D'_1
  44. Segment(D, D'_1) → t
  45. Ray(N, P) → f_1
  46. Dilate(B, If[12.5 < a ≤ 13.5, 13.5 - a, 0], C) → B'_2
  47. Vector(B, B'_2) → u_1
  48. Translate(B, u_1) → B'_3
  49. Translate(B'_1, u_1) → Q
  50. Segment(B'_3, Q) → n'
  51. Rotate(B'_3, If[13.5 < a ≤ 14.5, ((a - 13.5) * π / 2), π / 2], C) → R
  52. Rotate(Q, If[13.5 < a ≤ 14.5, ((a - 13.5) * π / 2), π / 2], C) → Q'
  53. Segment(R, Q') → n''
  54. Dilate(C, If[14.5 < a ≤ 15.5, 15.5 - a, 0], D) → C'_3
  55. Vector(C, C'_3) → v_1
  56. Translate(R, v_1) → R'
  57. Translate(Q', v_1) → Q''
  58. Segment(R', Q'') → n'''
  59. Intersect(q, r) → S
  60. Dilate(N, If[15.5 < a ≤ 16.5, 16.5 - a, 0], S) → N'
  61. Segment(N, N') → g_1
  62. Dilate(S, If[16.5 < a ≤ 17.5, 17.5 - a, 0], B) → S'
  63. Segment(S, S') → i_1
  64. Dilate(A, If[17.5 < a ≤ 18.5, 18.5 - a, 0], S) → A'_3
  65. Segment(A, A'_3) → j_1
  66. OrthogonalLine(B, g_1) → k_1
  67. Point(k_1) → E
  68. Angle(E, B, C) → α
  69. Dilate(B, If[18.5 < a ≤ 19.5, 19.5 - a, 0], S) → B'_4
  70. Vector(B, B'_4) → w_1
  71. Translate(p, w_1) → p'
  72. Rotate(p', If[18.5 < a ≤ 19.5, ((a - 18.5) * (π / 2 - α)), π / 2 - α], B'_4) → p''
  73. Line(C, S) → l_1
  74. Line(G, S) → m_1
  75. Point(g_1) → H
  76. Angle(G, S, H) → β
  77. Dilate(C, If[21 < a ≤ 22, 22 - a, 0], S) → C'_1
  78. Dilate(G, If[21 < a ≤ 22, 22 - a, 0], S) → G'
  79. Segment(C, C'_1) → n_1
  80. Segment(G, G') → p_1
  81. Rotate(p'', If[23 < a ≤ 24, ((-(a - 23)) * β), (-β)], S) → p'''

AI Drawing Prompt Reference

Draw a dynamic geometric problem demonstration, starting with a ray and a point to construct a quadrilateral, finding the midpoint of a segment. Perform scaling and rotation on the points to generate multiple points and segments. Draw a circle, find its intersection with a line, and use this point to draw a sector. Translate and rotate points to create vectors, and use them for transformations and combinations of shapes, ultimately forming a complex geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.