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Hidden Symmetry in Regular Polygons

Hidden Symmetry in Regular Polygons
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2019-04-25 00:42
PolygonTransformationGeometric Construction
This example demonstrates the hidden symmetry in regular polygons. It involves key mathematical concepts such as geometric transformations and symmetry. Using GeoGebra commands like Point, Polygon, LineBisector, etc., one can construct regular polygons and observe their symmetries. Suitable for middle school and above students or teachers.

GeoGebra Commands & Parameters

  1. Point(Circle[A, b_3]) → D
  2. Polygon(A, D, 6) → poly1, g, H, h, i, j, k, l, E, F, G
  3. LineBisector(k) → m
  4. LineBisector(j) → n
  5. LineBisector(i) → p
  6. Intersect(m, n) → CC
  7. Circle(G, a_3) → d_2
  8. Circle(F, a_3) → e
  9. Circle(E, a_3) → q
  10. Circle(D, a_3) → r
  11. Circle(A, a_3) → s_2
  12. Circle(H, a_3) → t_2
  13. Point(d_2) → P
  14. Point(e) → Q
  15. Point(t_2) → R
  16. Point(s_2) → S
  17. Point(r) → T
  18. Point(q) → U
  19. Line(P, G) → f_1
  20. Line(H, R) → g_1
  21. Line(S, A) → h_1
  22. Line(T, D) → i_1
  23. Line(U, E) → j_1
  24. Line(Q, F) → k_1
  25. Intersect(d_2, f_1, 2) → V
  26. Intersect(t_2, g_1, 2) → W
  27. Intersect(s_2, h_1, 2) → Z
  28. Intersect(r, i_1, 2) → A_1
  29. Intersect(q, j_1, 2) → B_1
  30. Intersect(e, k_1, 2) → C_1
  31. Segment(P, V) → l_1
  32. Segment(R, W) → m_1
  33. Segment(S, Z) → n_1
  34. Segment(T, A_1) → p_1
  35. Segment(U, B_1) → q_1
  36. Segment(C_1, Q) → r_1
  37. Segment(R, CC) → s
  38. Segment(P, CC) → t
  39. Segment(Q, CC) → a
  40. Segment(U, CC) → b
  41. Segment(T, CC) → c
  42. Segment(S, CC) → d
  43. Segment(W, CC) → s_1
  44. Segment(V, CC) → t_1
  45. Segment(C_1, CC) → a_1
  46. Segment(B_1, CC) → b_1
  47. Segment(A_1, CC) → c_1
  48. Segment(Z, CC) → d_1

AI Drawing Prompt Reference

Draw a regular hexagon and observe its hidden symmetry. Start by drawing a circle with point A as the center, then take a point D on the circle and draw a regular hexagon with D and A as vertices. Next, draw the diagonals of the hexagon and then find the perpendicular bisectors of these diagonals, with their intersection point CC being the center of the hexagon. Finally, draw line segments from each vertex to the center and their symmetric counterparts.
The prompt above was compiled from the original drawing commands and is for reference only.