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Dynamic Illustration of Two Regular Pentagons Sharing a Vertex

Dynamic Illustration of Two Regular Pentagons Sharing a Vertex
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2017-06-03 08:55
PolygonAngleTransformationGeometric Construction
This GeoGebra applet dynamically illustrates the geometric relationship between two regular pentagons sharing a common vertex. Key mathematical concepts include regular polygon properties, rotational symmetry, and angle-segment relationships. Construction utilizes commands such as Polygon, Rotate, Intersect, Angle, Vector, and Translate to create an interactive model. Suitable for high school geometry instruction or independent exploration of Euclidean geometry properties by math enthusiasts.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 5) → poly1, f, g, h, i, j, C, D, E
  2. Rotate(B, 144°, C) → B'
  3. Ray(C, B') → s
  4. Line(B, C) → k
  5. Circle(C, a) → c
  6. Intersect(c, s) → N,
  7. Intersect(c, k) → F, G
  8. CircleArc(C, F, N) → e
  9. Line(C, D) → l
  10. Intersect(c, l) → H, I
  11. CircleArc(C, F, I) → d
  12. Point(e) → J
  13. Polygon(C, J, 5) → poly2, m, n, p, q, r, K, L, M
  14. Angle(B, C, B') → α
  15. Segment(D, A) → t
  16. Segment(D, B) → b
  17. LineBisector(D, C) → f_1
  18. LineBisector(B, C) → g_1
  19. LineBisector(M, L) → h_1
  20. LineBisector(L, K) → i_1
  21. Segment(M, J) → j_1
  22. Segment(E, K) → k_1
  23. Segment(A, L) → l_1
  24. Intersect(k_1, l_1) → Q
  25. Circle(D, (0.2 * j_1)) → c_1
  26. Segment(D, E) → m_1
  27. Intersect(c_1, t, 1) → R
  28. Intersect(i, m_1) → S
  29. Intersect(c_1, m_1) → T,
  30. CircleSector(D, T, R) → d_1
  31. Dilate(E, If[0 < o ≤ 1, 1 - o, 0], K) → E'
  32. Segment(E, E') → n_1
  33. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], Q) → A'
  34. Segment(A, A') → p_1
  35. Dilate(Q, If[2 < o ≤ 3, 3 - o, 0], L) → Q'
  36. Segment(Q, Q') → q_1
  37. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], A) → D'
  38. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], B) → D'_1
  39. Segment(D, D') → r_1
  40. Segment(D, D'_1) → s_1
  41. Rotate(d_1, If[5 < o ≤ 6, (((o - 5) * 36) * π / 180), (36 * π / 180)], D) → e_1
  42. Rotate(e_1, If[7 < o ≤ 8, (((o - 7) * 36) * π / 180), (36 * π / 180)], D) → t_1
  43. Line(Q, s_1) → a_1
  44. Dilate(D, If[9 < o ≤ 10, 10 - o, 0], Q) → D'_2
  45. Vector(D, D'_2) → u
  46. Translate(t_1, u) → c_2
  47. Segment(D, E) → b_1
  48. Segment(E, A) → f_2
  49. Segment(A, B) → g_2
  50. Segment(B, C) → h_2
  51. Segment(C, D) → i_2
  52. Segment(C, M) → j_2
  53. Segment(M, L) → k_2
  54. Segment(L, K) → l_2
  55. Segment(K, J) → m_2
  56. Segment(J, C) → n_2
  57. Point(a_1) → O
  58. Angle(O, Q, K) → β
  59. Rotate(c_2, If[10 < o ≤ 11, ((o - 10) * β), β], Q) → d_2