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Dynamic Equal Area Demonstration in Regular Hexagon

Dynamic Equal Area Demonstration in Regular Hexagon
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2017-02-19 11:09
TrigonometryCoordinate GeometryIntegralParametric EquationGeometric Construction
In collections Perimeter & Area Perimeter & Area
This example dynamically illustrates equal area divisions within a regular hexagon. It involves core math concepts like parametric curves, real numbers, geometric transformations, and integral calculus. Construction uses commands such as Point, Polygon, Segment, Midpoint, Dilate, OrthogonalLine, Rotate, and Intersect. Suitable for high school and above mathematics education to understand geometric transformations and integral applications.

GeoGebra Commands & Parameters

  1. Point(xAxis) → B
  2. Polygon(A, B, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  3. Segment(A, D) → m
  4. Midpoint(m) → G
  5. Segment(G, B) → q_1
  6. Dilate(G, If[0 < o ≤ 1, 1 - o, 0], B) → G'
  7. Segment(G, G') → h_1
  8. OrthogonalLine(C, h_1) → p_1
  9. Segment(F, D) → l
  10. Segment(G, C) → n
  11. Polygon(A, G, B) → poly3, b, a_1, g_1
  12. Polygon(G, C, B) → poly4, b_1, g_2, c
  13. Segment(F, E) → p
  14. Segment(E, D) → q
  15. Segment(D, C) → r
  16. Segment(C, B) → s
  17. Segment(B, A) → t
  18. Segment(A, F) → e
  19. Polygon(A, G, F) → poly2, f_1, a, g_3
  20. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π / 3), ((-π)) / 3], G) → A'
  21. Rotate(G, If[1 < o ≤ 2, ((-(o - 1)) * π / 3), ((-π)) / 3], G) → G'_1
  22. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π / 3), ((-π)) / 3], G) → B'
  23. Polygon(A', G'_1, B') → poly3', b', a', g'_1
  24. Rotate(G, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → G'_2
  25. Rotate(C, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → C'
  26. Rotate(B, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → B'_1
  27. Midpoint(l) → H
  28. Dilate(E, If[3 < o ≤ 4, 4 - o, 0], F) → E'
  29. Segment(I, E') → j_1
  30. Intersect(j_1, l) → J
  31. Polygon(F, G, H) → poly8, h_3, f_3, g_5
  32. Polygon(G, H, D) → poly5, d, g_6, h_2
  33. Segment(G, A) → i_1
  34. Intersect(l, r) → K
  35. Segment(G, K) → k_1
  36. Segment(G, E) → l_1
  37. Intersect(l_1, l) → M
  38. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], F) → D'
  39. Segment(D, D') → n_1
  40. Intersect(p_1, q_1) → N
  41. Polygon(N, G, C) → poly6, c_1, n_2, g_4
  42. Polygon(N, C, B) → poly9, b_2, n_3, c_2
  43. Rotate(N, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → N'
  44. Rotate(G, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → G'_3
  45. Rotate(C, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → C'_1
  46. Polygon(N', G'_3, C'_1) → poly6', c'_1, n', g'_3
  47. Rotate(N, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → N'_1
  48. Rotate(C, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → C'_2
  49. Rotate(B, If[2 < o ≤ 3, ((2 * (o - 2)) * π / 3), (2 * π / 3)], G) → B'_2
  50. Polygon(N'_1, C'_2, B'_2) → poly9', b'_2, n'_1, c'_2
  51. Polygon(G, D, M) → poly7, m_1, g_7, d_1
  52. Polygon(D, M, E) → poly10, e_1, d_2, m_2
  53. Rotate(D, If[4 < o ≤ 5, ((o - 4) * π), π], M) → D'_1
  54. Rotate(M, If[4 < o ≤ 5, ((o - 4) * π), π], M) → M'
  55. Rotate(E, If[4 < o ≤ 5, ((o - 4) * π), π], M) → E'_1
  56. Polygon(D'_1, M', E'_1) → poly10', e'_1, d'_1, m'

AI Drawing Prompt Reference

Construct a regular hexagon and dynamically demonstrate equal-area divisions within it. Start by drawing a segment from one vertex to the midpoint of the opposite side, then take its midpoint as a rotation center. Use a slider parameter to control rotations and dilations, dividing the hexagon into several triangular regions that maintain equal areas as the parameter changes. Include auxiliary segments and intersection points to illustrate the area relationships. The figure shows a symmetric geometric construction suitable for educational demonstrations.
The prompt above was compiled from the original drawing commands and is for reference only.