积极 AI Logo 积极 AI

纸折叠三等分角度

纸折叠三等分角度
几何画板及绘图内容加载中，请稍等……
2018-03-19 10:01
测试改名0角变换几何作图滑块动画
①本范例演示通过纸折叠实现角度三等分的几何作图方法；②涉及核心数学概念包括三角形、角度、几何变换与作图原理；③构造过程中运用了点、射线、圆、旋转、角度测量、交点、角平分线、多边形、距离计算、扇形、缩放、向量、平移、直线、线段等 GeoGebra 命令与对象；④适合高中数学教学或几何爱好者探索几何作图技巧。

GeoGebra 命令及参数

  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