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Regular Octagon Problem Dynamic Demo

Regular Octagon Problem Dynamic Demo
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2019-01-28 01:24
PolygonCoordinate GeometryLimitTransformationGeometric Construction
This example dynamically demonstrates a geometric problem posed by Catriona Shearer, showcasing interactive illustrations of polygonal relationships. Key math concepts include coordinate geometry, geometric transformations, limits, and curve sketching. Construction highlights GeoGebra commands like Polygon, Line, Intersect, Segment, Rotate, Circle, Dilate, Vector, Translate, OrthogonalLine, Point, and Angle. Suitable for high school and above, aiding in understanding complex geometric solutions and dynamic changes.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 8) → poly1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  2. Polygon(F, G, 4) → poly2, n, p, q, r, I, J
  3. Polygon(B, C, 4) → poly3, s, t, a, b, K, L
  4. Line(L, K) → c
  5. Line(J, I) → d
  6. Intersect(d, i) → M
  7. Intersect(c, i) → N
  8. Intersect(c, m) → O
  9. Intersect(d, m) → P
  10. Polygon(O, N, M, P) → q1, o, n_1, m_1, p_1
  11. Polygon(G, I, P, H) → q2, g_1, i_1, p_2, h_1
  12. Polygon(O, A, B, L) → q3, o_1, a_1, b_1, l_1
  13. Polygon(K, N, D, C) → q4, k_1, n_2, d_1, c_1
  14. Polygon(M, E, F, J) → q5, m_2, e, f_1, j_1
  15. Segment(G, H) → q_1
  16. Segment(H, A) → r_1
  17. Segment(A, B) → s_1
  18. Segment(B, C) → t_1
  19. Segment(C, D) → e_1
  20. Segment(D, E) → f_2
  21. Segment(E, F) → g_2
  22. Segment(F, G) → h_2
  23. Segment(O, N) → i_2
  24. Segment(M, P) → j_2
  25. Segment(C, K) → k_2
  26. Segment(L, B) → l_2
  27. Segment(J, F) → q_2
  28. Segment(I, G) → r_2
  29. Rotate(poly2, If[0 < u ≤ 1, ((u * 3) * π / 4), (3 * π / 4)], F) → poly2'
  30. Rotate(J, If[0 < u ≤ 1, ((u * 3) * π / 4), (3 * π / 4)], F) → J'
  31. Rotate(I, If[0 < u ≤ 1, ((u * 3) * π / 4), (3 * π / 4)], F) → I'
  32. Line(J', g_2) → s_2
  33. Segment(M, N) → t_2
  34. Line(D, E) → a_2
  35. Circle(F, j_2) → c_2
  36. Intersect(c_2, s_2, 1) → R
  37. Dilate(J', If[2 < u ≤ 3, 3 - u, 0], R) → J''
  38. Vector(J', J'') → v
  39. Translate(I', v) → I''
  40. Translate(F, v) → F'
  41. Polygon(F, E, I'', J'') → q6, f_3, e_2, i'', j''
  42. Segment(J'', F) → b_2
  43. Segment(E, I'') → d_2
  44. Segment(I'', J'') → g_3
  45. Rotate(G, If[0 < u ≤ 1, ((u * 3) * π / 4), (3 * π / 4)], F) → G'
  46. Segment(G', I') → h_3
  47. Segment(I', J') → i_3
  48. Segment(J', F) → j_3
  49. Segment(F, G') → k_3
  50. OrthogonalLine(E, d_2) → l_3
  51. Intersect(l_3, b_2) → S
  52. Dilate(F, If[4 < u ≤ 5, 5 - u, 0], S) → F'_1
  53. Vector(F, F'_1) → w
  54. Translate(J'', w) → J'''
  55. Polygon(F'_1, E, I'', J''') → q7, f'_1, e_3, i''_1, j'''
  56. Segment(E, F'_1) → m_3
  57. Segment(F'_1, J''') → n_3
  58. Segment(J''', I'') → p_3
  59. Dilate(E, If[6 < u ≤ 7, 7 - u, 0], O) → E'
  60. Vector(E, E') → u_1
  61. Translate(F'_1, u_1) → Q
  62. Translate(E, u_1) → E'_1
  63. Translate(I'', u_1) → I'''
  64. Translate(J''', u_1) → T
  65. Polygon(Q, E'_1, I''', T) → q7', q_3, e'_1, i''', t_3
  66. Line(O, m_3) → r_3
  67. Point(r_3) → U
  68. Angle(U, O, H) → α
  69. Rotate(Q, If[6 < u ≤ 7, ((u - 6) * α), α], E'_1) → Q'
  70. Rotate(E'_1, If[6 < u ≤ 7, ((u - 6) * α), α], E'_1) → V
  71. Rotate(I''', If[6 < u ≤ 7, ((u - 6) * α), α], E'_1) → W
  72. Rotate(T, If[6 < u ≤ 7, ((u - 6) * α), α], E'_1) → T'
  73. Polygon(Q', V, W, T') → q7'', q', v_1, w_1, t'
  74. Line(J', g_2) → s_3
  75. Line(F, J'') → a_3

AI Drawing Prompt Reference

Draw a regular octagon and dynamically demonstrate its internal geometric relationships. First, construct the octagon, then create several quadrilaterals and triangles inside it, connecting with intersection points and line segments. Next, perform rotations and dilations on some parts to show dynamic transformations. Finally, add some auxiliary line segments and circles to enrich the figure.
The prompt above was compiled from the original drawing commands and is for reference only.