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Paper Folding Angle Trisection

Paper Folding Angle Trisection
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2018-03-19 10:01
TriangleAngleTransformationGeometric ConstructionSlider Animation
This example demonstrates the geometric construction of angle trisection using paper folding. It involves core mathematical concepts such as triangles, angles, geometric transformations, and construction principles. The construction utilizes GeoGebra commands and objects including points, rays, circles, rotations, angle measurements, intersections, angle bisectors, polygons, distance calculations, sectors, dilations, vectors, translations, lines, and segments. It is suitable for high school mathematics instruction or exploration by geometry enthusiasts.

GeoGebra Commands & Parameters

  1. Point(yAxis) → C
  2. Point(yAxis) → A
  3. Ray(A, C) → g
  4. Point(g) → D
  5. Circle(A, (0.5 * Distance[A, D])) → d_3
  6. Point(xAxis) → B
  7. Rotate(B, (a * π / 180), A) → B'
  8. Ray(A, B') → h
  9. Ray(A, B) → f
  10. Angle(B, A, B') → α
  11. Rotate(B, α / 3, A) → B'_1
  12. Angle(B, A, B'_1) → β
  13. Rotate(B'_1, α / 3, A) → E
  14. Angle(B'_1, A, E) → γ
  15. Ray(A, B'_1) → k
  16. Ray(A, E) → l
  17. Intersect(i, k) → F
  18. Intersect(g, yAxis) → G
  19. Intersect(g, yAxis) → H
  20. Intersect(j, g) → I
  21. Intersect(h, j) → J
  22. LineBisector(A, F) → m
  23. Intersect(m, f) → K
  24. Intersect(m, g) → L
  25. Polygon(L, A, K) → t1, k_1, l_1, a_1
  26. Polygon(K, F, L) → t2, l_2, k_2, f_1
  27. Intersect(k_2, h) → M
  28. Intersect(j, m) → N
  29. Distance(I, N) → distanceIN
  30. Distance(N, M) → distanceNM
  31. Distance(A, D) → c
  32. Circle(A, (0.3 * c)) → d
  33. Intersect(g, k_1) → O
  34. Intersect(g, k_1) → P
  35. Intersect(g, yAxis) → Q
  36. Intersect(g, yAxis) → R
  37. Intersect(d, g) → S,
  38. Rotate(S, If[0 < b ≤ 1, (π / 2 * b), π / 2], A) → S'
  39. CircleSector(A, S, S') → e
  40. Dilate(A, If[2 < b ≤ 3, 3 - b, 0], D) → A'
  41. Vector(A, A') → u
  42. Translate(e, u) → e'
  43. Line(A, K) → n
  44. Segment(A, A') → p
  45. Dilate(D, If[4 < b ≤ 5, 5 - b, 0], I) → D'
  46. Vector(D, D') → v
  47. Translate(e', v) → e''
  48. Segment(D, D') → q
  49. Rotate(A, If[5 < b ≤ 6, ((b - 5) * π), π], D) → A'_1
  50. Rotate(A', If[5 < b ≤ 6, ((b - 5) * π), π], D) → A''
  51. Segment(A'_1, A'') → p'
  52. Line(A, D) → r
  53. Circle(I, 2) → s
  54. Intersect(s, r, 2) → T
  55. Ray(I, T) → t
  56. Segment(D, A') → g_1
  57. Segment(D, A) → h_1
  58. Intersect(i, m) → U
  59. Intersect(l, j) → V
  60. Line(L, U) → j_1
  61. Dilate(L, If[6 < b ≤ 7, 7 - b, 0], N) → L'
  62. Segment(L, L') → i_1
  63. Distance(I, i_1) → distanceIi1
  64. Distance(M, i_1) → distanceMi1
  65. Segment(A, K) → m_1
  66. Dilate(N, If[7 < b ≤ 8, 8 - b, 0], U) → N'
  67. Segment(N, N') → n_1
  68. Dilate(U, If[8 < b ≤ 9, 9 - b, 0], K) → U'
  69. Segment(U, U') → p_1
  70. Circle(K, 2) → d_1
  71. Circle(N, 2) → e_1
  72. Circle(A, 2) → q_1
  73. Intersect(q_1, n) → W, Z
  74. Intersect(d_1, n) → A_1, B_1
  75. Intersect(e_1, j) → C_1, D_1
  76. Ray(A, W) → r_1
  77. Ray(K, B_1) → s_1
  78. Ray(N, D_1) → t_1
  79. Circle(I, 2) → c_2
  80. Intersect(c_2, j) → E_1, F_1
  81. Ray(I, E_1) → f_2
  82. Segment(I, N) → g_2
  83. Dilate(I, If[9 < b ≤ 10, 10 - b, 0], N) → I'
  84. Dilate(A, If[9 < b ≤ 10, 10 - b, 0], K) → A'_2
  85. Segment(I, I') → h_2
  86. Segment(A, A'_2) → i_2
  87. Angle(M, N, I) → δ
  88. Angle(F, K, A) → ε
  89. Rotate(I, If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], N) → I'_1
  90. Rotate(I', If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], N) → I''
  91. Segment(I'_1, I'') → j_2
  92. Rotate(A, If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], K) → A'_3
  93. Rotate(A'_2, If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], K) → G_1
  94. Segment(A'_3, G_1) → m_2
  95. Rotate(A, If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], K) → A'_4
  96. Rotate(A'_2, If[10 < b ≤ 11, ((-(b - 10)) * ε), (-ε)], K) → H_1
  97. Angle(F, L, A) → ζ
  98. Circle(L, 2) → d_2
  99. Intersect(d_2, t) → I_1, J_1
  100. Segment(I, L) → n_2
  101. Ray(L, J_1) → p_2
  102. Rotate(I, If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → I'_2
  103. Rotate(L, If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → L'_1
  104. Segment(I'_2, L'_1) → q_2
  105. Rotate(A'_1, If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → K_1
  106. Rotate(A'', If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → A'''
  107. Segment(K_1, A''') → p''
  108. Rotate(A, If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → A'_5
  109. Rotate(A', If[11 < b ≤ 12, ((b - 11) * ζ), ζ], L) → A''_1
  110. Segment(A'_5, A''_1) → p'_1
  111. Intersect(d, y = y(A)) → O_1, P_1
  112. Intersect(d_3, h) → L_1,
  113. CircleSector(A, P_1, L_1) → e_2
  114. Line(A, U) → r_2
  115. Line(A, F) → s_2
  116. Circle(U, 2) → t_2
  117. Circle(F, 2) → c_3
  118. Intersect(t_2, r_2, 2) → M_1
  119. Intersect(c_3, s_2) → N_1, Q_1
  120. Ray(U, M_1) → a_2
  121. Ray(F, Q_1) → b_2
  122. Dilate(A, If[14 < b ≤ 15, 15 - b, 0], U) → A'_6
  123. Dilate(A, If[15 < b ≤ 16, 16 - b, 0], F) → A'_7
  124. Segment(A, A'_6) → g_3
  125. Segment(A, A'_7) → f_3
  126. Rotate(L_1, If[17 < b ≤ 18, ((-(b - 17)) * α / 3), ((-α)) / 3], A) → R_1
  127. CircleSector(A, R_1, L_1) → e_3
  128. Rotate(e_3, If[19 < b ≤ 20, ((-(b - 19)) * α / 3), ((-α)) / 3], A) → h_3
  129. Rotate(h_3, If[21 < b ≤ 22, ((-(b - 21)) * α / 3), ((-α)) / 3], A) → k_3