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Equal Triangular Areas in a Regular Pentagon

Equal Triangular Areas in a Regular Pentagon
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2017-02-14 08:14
TrianglePolygonTransformationGeometric Construction
This example dynamically illustrates equal triangular areas within a regular pentagon. Key concepts include polygons, triangles, and rotation. Using GeoGebra commands such as Rotate, Ray, Point, Segment, Circle, and Intersect, the distribution of triangle areas can be visually understood. Suitable for middle school and above students to learn geometric transformations and area calculations.

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. Segment(J, B) → l_1
  72. Segment(C, J) → m_1
  73. Point(b'') → N
  74. Segment(K, C) → n_1
  75. Segment(N, C) → p_1
  76. Segment(N, K) → q_1
  77. OrthogonalLine(N, n_1) → r_1
  78. Intersect(r_1, n_1) → O
  79. OrthogonalLine(B, m_1) → s_1
  80. Intersect(s_1, m_1) → P
  81. Polygon(B, C, J) → poly2, j_2, b_2, c_3
  82. Segment(N, O) → t_1
  83. Segment(B, P) → a_1
  84. Dilate(B, If[15 < o ≤ 16, 16 - o, 0], J) → B'_3
  85. Segment(B, B'_3) → f_2
  86. Dilate(C, If[15 < o ≤ 16, 16 - o, 0], J) → C'_2
  87. Segment(C, C'_2) → h_2
  88. Angle(J, P, B) → ε
  89. Rotate(B, If[16 < o ≤ 17, ((o - 16) * α / 3), α / 3], J) → B'_4
  90. Rotate(B'_3, If[16 < o ≤ 17, ((o - 16) * α / 3), α / 3], J) → Q
  91. Segment(B'_4, Q) → i_2
  92. Dilate(B, If[17 < o ≤ 18, 18 - o, 0], P) → B'_5
  93. Segment(B, B'_5) → l_2
  94. Rotate(B'_4, If[18 < o ≤ 19, ((o - 18) * α / 3), α / 3], C) → R
  95. Rotate(Q, If[18 < o ≤ 19, ((o - 18) * α / 3), α / 3], C) → Q'
  96. Segment(R, Q') → m_2
  97. Dilate(K, If[19 < o ≤ 20, 20 - o, 0], N) → K'_1
  98. Dilate(C, If[19 < o ≤ 20, 20 - o, 0], N) → C'_3
  99. Segment(K, K'_1) → n_2
  100. Segment(C, C'_3) → p_2
  101. Polygon(K, C, N) → poly3, n_3, k_3, c_4
  102. Line(O, l_2) → q_2
  103. Circle(O, 5) → d_2
  104. Intersect(d_2, q_2, 2) → S
  105. Angle(S, O, N) → ζ
  106. Dilate(N, If[20.5 < o ≤ 21.5, 21.5 - o, 0], O) → N'
  107. Segment(N, N') → r_2
  108. Angle(N, O, K) → η
  109. Dilate(B, If[21.5 < o ≤ 22.5, 22.5 - o, 0], O) → B'_6
  110. Vector(B, B'_6) → u_1
  111. Translate(B, u_1) → B'_7
  112. Translate(B'_5, u_1) → T
  113. Segment(B'_7, T) → s_2
  114. Rotate(B'_7, If[22.5 < o ≤ 23.5, ((o - 22.5) * α / 3), α / 3], O) → U
  115. Rotate(T, If[22.5 < o ≤ 23.5, ((o - 22.5) * α / 3), α / 3], O) → T'
  116. Segment(U, T') → t_2