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Dynamic Demonstration of Concyclic Points from Pentagons and Triangle

Dynamic Demonstration of Concyclic Points from Pentagons and Triangle
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2017-04-24 10:12
CirclePolygonVectorTrigonometryGeometric Construction
This example dynamically illustrates how two regular pentagons and one equilateral triangle can form four concyclic points. It involves core mathematical concepts such as regular polygons, concyclic points, rotation, similarity, and vectors. The construction utilizes GeoGebra commands and objects like Rotate, Circle, Intersect, and Vector as detailed in the summary. Suitable for high school and above students to explore complex geometric properties and constructions.

GeoGebra Commands & Parameters

  1. Rotate(A, 108°, B) → A'
  2. Ray(B, A') → g
  3. Ray(B, A) → f
  4. Point(f) → C
  5. Segment(C, B) → n
  6. Circle(C, n) → d_1
  7. Rotate(B, (-(108°)), C) → B'
  8. Ray(C, B') → h
  9. Intersect(d_1, h, 1) → G
  10. Circle(G, n) → e_1
  11. Point(h) → D
  12. Rotate(C, (-(108°)), D) → C'
  13. Ray(D, C') → i
  14. Point(i) → E
  15. Rotate(D, (-(108°)), E) → D'
  16. Ray(E, D') → j
  17. Intersect(g, j) → F
  18. Segment(F, E) → k
  19. Segment(E, D) → l
  20. Line(G, l) → g_1
  21. Intersect(e_1, g_1, 1) → J
  22. Line(J, k) → h_1
  23. Intersect(h_1, g) → K
  24. Angle(A, B, A') → α
  25. Angle(B', C, B) → β
  26. Angle(C', D, C) → γ
  27. Angle(D', E, D) → δ
  28. Segment(D, C) → m
  29. Segment(B, F) → p
  30. Min(list) → a
  31. Circle(B, (0.2 * n)) → q
  32. Intersect(q, p) → H,
  33. Intersect(q, n) → I,
  34. CircleSector(B, I, H) → r
  35. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], C) → B'_1
  36. Vector(B, B'_1) → u
  37. Translate(r, u) → r'
  38. Rotate(r', If[1 < o ≤ 2, (((o - 1) * 2) * π / 5), (2 * π / 5)], C) → r''
  39. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], G) → C'_1
  40. Vector(C, C'_1) → v
  41. Translate(r'', v) → r'''
  42. Rotate(r''', If[3 < o ≤ 4, (((o - 3) * 2) * π / 5), (2 * π / 5)], G) → s
  43. Dilate(G, If[4 < o ≤ 5, 5 - o, 0], J) → D'_1
  44. Vector(G, D'_1) → w
  45. Translate(s, w) → s'
  46. Rotate(s', If[5 < o ≤ 6, (((o - 5) * 2) * π / 5), (2 * π / 5)], J) → s''
  47. Dilate(J, If[6 < o ≤ 7, 7 - o, 0], K) → E'
  48. Vector(J, E') → t
  49. Translate(s'', t) → s'''
  50. Circle(J, n) → k_1
  51. Rotate(s''', If[7 < o ≤ 8, (((o - 7) * 2) * π / 5), (2 * π / 5)], K) → c_1
  52. Segment(B, B'_1) → b
  53. Segment(C, C'_1) → c
  54. Segment(G, D'_1) → d
  55. Segment(J, E') → e
  56. Dilate(K, If[8 < o ≤ 9, 9 - o, 0], B) → K'
  57. Segment(K, K') → f_1
  58. Polygon(B, C, G, J, K) → poly1, b_1, c_2, g_2, j_1, k_2
  59. Rotate(B, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → B'_2
  60. Rotate(B'_1, If[9 < o ≤ 10, ((-(o - 9)) * α), (-α)], C) → L
  61. Segment(B'_2, L) → b'
  62. Rotate(B'_2, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G) → M
  63. Rotate(L, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G) → L'
  64. Segment(M, L') → b''
  65. Rotate(M, If[11 < o ≤ 12, ((-(o - 11)) * α), (-α)], J) → M'
  66. Rotate(L', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], J) → L''
  67. Segment(M', L'') → b'''
  68. Rotate(M', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → M''
  69. Rotate(L'', If[12 < o ≤ 13, ((-(o - 12)) * α), (-α)], K) → L'''
  70. Segment(M'', L''') → i_1
  71. Rotate(B, If[13 < o ≤ 14, ((-(o - 13)) * π / 3), ((-π)) / 3], B) → B'_3
  72. Rotate(B'_1, If[13 < o ≤ 14, ((-(o - 13)) * π / 3), ((-π)) / 3], B) → N
  73. Segment(B'_3, N) → b'_1
  74. Rotate(C, ((-π)) / 3, B) → C'_2
  75. Segment(J, C'_2) → l_1
  76. Rotate(B, If[14 < o ≤ 15, ((o - 14) * π / 3), π / 3], C) → B'_4
  77. Rotate(B'_1, If[14 < o ≤ 15, ((o - 14) * π / 3), π / 3], C) → O
  78. Segment(B'_4, O) → b'_2
  79. Dilate(J, If[15 < o ≤ 16, 16 - o, 0], C'_2) → J'
  80. Segment(J, J') → m_1
  81. Polygon(B, C, C'_2) → poly2, c'_2, b_2, c_3
  82. Dilate(B, If[17 < o ≤ 18, 18 - o, 0], C'_2) → B'_5
  83. Vector(B, B'_5) → u_1
  84. Translate(B, u_1) → B'_6
  85. Translate(C, u_1) → C'_3
  86. Translate(G, u_1) → G'
  87. Translate(J, u_1) → J'_1
  88. Translate(K, u_1) → K'_1
  89. Polygon(B'_6, C'_3, G', J'_1, K'_1) → poly1', b'_6, c'_3, g', j'_1, k'_1
  90. Rotate(B'_6, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → P
  91. Rotate(C'_3, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → Q
  92. Rotate(G', If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → G''
  93. Rotate(J'_1, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → R
  94. Rotate(K'_1, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], C'_2) → S
  95. Polygon(P, Q, G'', R, S) → poly1'', p_1, q_1, g'', r_1, s_1
  96. Segment(C'_2, J) → n_1
  97. Distance(C'_2, b) → a_1
  98. Circle(C'_2, a_1) → t_1
  99. Distance(C'_2, J) → o_1
  100. Circle(C'_2, b) → d_2
  101. Intersect(m_1, n_1) → T
  102. Intersect(d_2, n_1) → U,
  103. Dilate(U, If[19.5 < o ≤ 20.5, 20.5 - o, 0], J) → U'
  104. Polygon(C'_2, U', 5) → poly3, f_2, h_2, i_2, j_2, l_2, V, W, Z
  105. LineBisector(W, Z) → m_2
  106. LineBisector(E', W) → n_2
  107. Intersect(n_2, m_2) → A_1
  108. Circle(A_1, W) → e_2
  109. Dilate(A_1, If[20.5 < o ≤ 21.5, 21.5 - o, 0], W) → B_1
  110. Dilate(A_1, If[21 < o ≤ 22, 22 - o, 0], W) → C_1
  111. Segment(A_1, C_1) → p_2
  112. Angle(E', A_1, WW) → ε
  113. Angle(B, A_1, E') → ζ
  114. Angle(Z, A_1, B) → μ
  115. Angle(WW, A_1, Z) → θ
  116. Rotate(WW, If[22 < o ≤ 23, ((-(o - 22)) * ε), (-ε)], A_1) → WW'
  117. CircleArc(A_1, WW', WW) → q_2
  118. Rotate(E', If[23 < o ≤ 24, ((-(o - 23)) * ζ), (-ζ)], A_1) → E''
  119. CircleArc(A_1, E'', E') → r_2
  120. Rotate(B, If[24 < o ≤ 25, ((-(o - 24)) * μ), (-μ)], A_1) → B'_7
  121. CircleArc(A_1, B'_7, B) → s_2
  122. Rotate(Z, If[25 < o ≤ 26, ((-(o - 25)) * θ), (-θ)], A_1) → Z'
  123. CircleArc(A_1, Z', Z) → t_2

AI Drawing Prompt Reference

Draw two regular pentagons and an equilateral triangle to demonstrate how they form four concyclic points. Rotate one pentagon's vertex by 108 degrees, then draw rays from the center to the vertices, intersecting to form circles. Repeat this rotation and intersection process for the pentagon and triangle, highlighting the formation of four concyclic points. Use vectors to represent rotations and circles and intersections to emphasize the concyclic relationship, dynamically illustrating geometric properties.
The prompt above was compiled from the original drawing commands and is for reference only.