MathPlotAI Logo MathPlotAI

Dynamic Relationship of a Regular Pentagon and Its Circumcircle

Dynamic Relationship of a Regular Pentagon and Its Circumcircle
Loading the math board and drawing, please wait…
2017-06-08 02:03
CirclePolygonAngleVectorGeometric Construction
This example demonstrates the dynamic and modifiable relationship between a regular pentagon, its circumcircle, and several other segments. Key math concepts include polygons, circles, angles, vectors, and geometric transformations. The construction utilizes GeoGebra commands such as Rotate, Ray, Point, Segment, Circle, Intersect, Line, Angle, Min, CircleSector, Dilate, Vector, Translate, Polygon, LineBisector, OrthogonalLine, CircleArc, and Midpoint. It is suitable for high school and above students to explore complex geometric interactions and transformations.

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_1
  10. Circle(G_1, n) → e_1
  11. Point(h) → D_1
  12. Rotate(C, (-(108°)), D_1) → C'
  13. Ray(D_1, C') → i
  14. Point(i) → E
  15. Rotate(D_1, (-(108°)), E) → D'
  16. Ray(E, D') → j
  17. Intersect(g, j) → F_1
  18. Segment(F_1, E) → k
  19. Segment(E, D_1) → l
  20. Line(G_1, 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_1, C) → γ
  27. Angle(D', E, D_1) → δ
  28. Segment(D_1, C) → m
  29. Segment(B, F_1) → 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_1) → 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_1) → s
  43. Dilate(G_1, If[4 < o ≤ 5, 5 - o, 0], J) → D'_1
  44. Vector(G_1, 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_1, 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_1, 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_1) → M
  63. Rotate(L, If[10 < o ≤ 11, ((-(o - 10)) * α), (-α)], G_1) → 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. LineBisector(B, C) → l_1
  72. LineBisector(G_1, J) → m_1
  73. Intersect(m_1, l_1) → N
  74. Circle(N, J) → p_1
  75. Intersect(p_1, m_1) → D, P
  76. Line(N, G_1) → n_1
  77. Line(J, G_1) → q_1
  78. Intersect(q_1, m_1) → F
  79. OrthogonalLine(F, n_1) → r_1
  80. Intersect(r_1, n_1) → G
  81. Line(J, D) → s_1
  82. Intersect(n_1, s_1) → O
  83. Angle(F, G, N) → ε
  84. Line(N, G_1) → t_1
  85. Intersect(p_1, t_1) → Q, R
  86. Rotate(J, If[13 < o ≤ 14, (((o - 13) * 288) * π / 180), (288 * π / 180)], N) → J'
  87. CircleArc(N, J, J') → d_2
  88. Rotate(G_1, If[14 < o ≤ 15, (((o - 14) * 36) * π / 180), (36 * π / 180)], N) → S
  89. CircleArc(N, G_1, S) → e_2
  90. Rotate(D, If[15 < o ≤ 16, (((o - 15) * 36) * π / 180), (36 * π / 180)], N) → D'_2
  91. CircleArc(N, D, D'_2) → f_2
  92. CircleArc(N, J, K) → h_2
  93. CircleArc(N, K, B) → p_2
  94. CircleArc(N, B, C) → q_2
  95. CircleArc(N, C, G_1) → r_2
  96. CircleArc(N, G_1, J) → s_2
  97. Rotate(e_2, If[18 < o ≤ 19, (((o - 18) * 36) * π / 180), (36 * π / 180)], N) → t_2
  98. Dilate(N, If[19 < o ≤ 20, 20 - o, 0], F) → N'
  99. Segment(N, N') → a_1
  100. Dilate(F, If[20 < o ≤ 21, 21 - o, 0], D) → F'
  101. Segment(F, F') → i_2
  102. Dilate(F, If[21 < o ≤ 22, 22 - o, 0], G) → F'_1
  103. Dilate(N, If[21 < o ≤ 22, 22 - o, 0], G) → N'_1
  104. Segment(N, N'_1) → j_2
  105. Segment(F, F'_1) → l_2
  106. Dilate(G, If[22 < o ≤ 23, 23 - o, 0], O) → G'
  107. Dilate(J, If[22 < o ≤ 23, 23 - o, 0], O) → J'_1
  108. Segment(J, J'_1) → m_2
  109. Segment(G, G') → n_2
  110. Dilate(N, If[22 < o ≤ 23, 23 - o, 0], Q) → N'_2
  111. Segment(N, N'_2) → a_2
  112. Midpoint(G, N) → T
  113. Rotate(G, If[24.5 < o ≤ 25.5, ((-(o - 24.5)) * π), (-π)], T) → G'_1
  114. Rotate(G', If[24.5 < o ≤ 25.5, ((-(o - 24.5)) * π), (-π)], T) → G''
  115. Segment(G'_1, G'') → b_2