无字证明：毕达哥拉斯定理的六边形拼图

GeoGebra 命令及参数

1. Point(yAxis) → A
2. Point(yAxis) → B
3. Semicircle(A, B) → c
4. Point(c) → C
5. Polygon(A, C, 6) → poly1, f, G, g, h, i, j, k, D, E, F
6. Polygon(C, B, 6) → poly2, l, K, m, n, p, q, r, H, I, J
7. Polygon(B, A, 6) → poly3, s, O, t, a, b, d, e, L, M, N
8. Centroid(poly2) → P
9. Midpoint(B, C) → Q
10. Midpoint(C, K) → R
11. Midpoint(K, J) → S
12. Midpoint(J, I) → T
13. Midpoint(I, H) → U
14. Midpoint(H, B) → V
15. Angle(C, B, A) → α
16. Mirror(A, C) → A'
17. Line(B, A') → f_1
18. Line(Q, P) → g_1
19. Intersect(f_1, g_1) → W
20. Polygon(P, R, C, W) → q1, p_1, r_1, c_1, w
21. Rotate(P, 60°, P) → P'
22. Rotate(R, 60°, P) → R'
23. Rotate(C, 60°, P) → C'
24. Rotate(W, 60°, P) → W'
25. Polygon(P', R', C', W') → q1', p', r', c', w'
26. Rotate(P', 60°, P) → P''
27. Rotate(R', 60°, P) → R''
28. Rotate(C', 60°, P) → C''
29. Rotate(W', 60°, P) → W''
30. Polygon(P'', R'', C'', W'') → q1'', p'', r'', c'', w''
31. Rotate(P'', 60°, P) → P'''
32. Rotate(R'', 60°, P) → R'''
33. Rotate(C'', 60°, P) → C'''
34. Rotate(W'', 60°, P) → W'''
35. Polygon(P''', R''', C''', W''') → q1''', p''', r''', c''', w'''
36. Rotate(P''', 60°, P) → Z
37. Rotate(R''', 60°, P) → A_1
38. Rotate(C''', 60°, P) → B_1
39. Rotate(W''', 60°, P) → C_1
40. Polygon(Z, A_1, B_1, C_1) → q2, z_1, a_1, b_1, h_1
41. Rotate(Z, 60°, P) → Z'
42. Rotate(A_1, 60°, P) → D_1
43. Rotate(B_1, 60°, P) → E_1
44. Rotate(C_1, 60°, P) → F_1
45. Polygon(Z', D_1, E_1, F_1) → q2', z', d_1, e_1, i_1
46. Polygon(W, R', C) → t1, c_2, w_1, j_1
47. Polygon(W', V, B) → t2, b_2, k_1, v
48. Polygon(W'', H, U) → t3, u, l_1, h_2
49. Polygon(I, W''', T) → t4, t_1, i_2, m_1
50. Polygon(J, C_1, S) → t5, s_1, j_2, n_1
51. Polygon(K, F_1, R) → t6, r_2, k_2, q_1
52. If(0 < o ≤ 1, Polygon[Translate[P, Vector[C, Dilate[C, 1 - o, M]]], Translate[R, Vector[C, Dilate[C, 1 - o, M]]], Translate[C, Vector[C, Dilate[C, 1 - o, M]]], Translate[W, Vector[C, Dilate[C, 1 - o, M]]]], Polygon[Translate[P, Vector[C, M]], Translate[R, Vector[C, M]], Translate[C, Vector[C, M]], Translate[W, Vector[C, M]]]) → q3
53. If(2 < o ≤ 3, Polygon[Translate[P', Vector[B, Dilate[B, 3 - o, N]]], Translate[R', Vector[B, Dilate[B, 3 - o, N]]], Translate[C', Vector[B, Dilate[B, 3 - o, N]]], Translate[W', Vector[B, Dilate[B, 3 - o, N]]]], Polygon[Translate[P', Vector[B, N]], Translate[R', Vector[B, N]], Translate[C', Vector[B, N]], Translate[W', Vector[B, N]]]) → q4
54. If(4 < o ≤ 5, Polygon[Translate[P'', Vector[H, Dilate[H, 5 - o, O]]], Translate[R'', Vector[H, Dilate[H, 5 - o, O]]], Translate[C'', Vector[H, Dilate[H, 5 - o, O]]], Translate[W'', Vector[H, Dilate[H, 5 - o, O]]]], Polygon[Translate[P'', Vector[H, O]], Translate[R'', Vector[H, O]], Translate[C'', Vector[H, O]], Translate[W'', Vector[H, O]]]) → q5
55. If(6 < o ≤ 7, Polygon[Translate[P''', Vector[I, Dilate[I, 7 - o, B]]], Translate[R''', Vector[I, Dilate[I, 7 - o, B]]], Translate[C''', Vector[I, Dilate[I, 7 - o, B]]], Translate[W''', Vector[I, Dilate[I, 7 - o, B]]]], Polygon[Translate[P''', Vector[I, B]], Translate[R''', Vector[I, B]], Translate[C''', Vector[I, B]], Translate[W''', Vector[I, B]]]) → q6
56. If(8 < o ≤ 9, Polygon[Translate[Z, Vector[J, Dilate[J, 9 - o, A]]], Translate[A_1, Vector[J, Dilate[J, 9 - o, A]]], Translate[B_1, Vector[J, Dilate[J, 9 - o, A]]], Translate[C_1, Vector[J, Dilate[J, 9 - o, A]]]], Polygon[Translate[Z, Vector[J, A]], Translate[A_1, Vector[J, A]], Translate[B_1, Vector[J, A]], Translate[C_1, Vector[J, A]]]) → q7
57. If(10 < o ≤ 11, Polygon[Translate[Z', Vector[K, Dilate[K, 11 - o, L]]], Translate[D_1, Vector[K, Dilate[K, 11 - o, L]]], Translate[E_1, Vector[K, Dilate[K, 11 - o, L]]], Translate[F_1, Vector[K, Dilate[K, 11 - o, L]]]], Polygon[Translate[Z', Vector[K, L]], Translate[D_1, Vector[K, L]], Translate[E_1, Vector[K, L]], Translate[F_1, Vector[K, L]]]) → q8
58. If(12 < o ≤ 13, Polygon[Translate[W, Vector[C, Dilate[C, 13 - o, A]]], Translate[R', Vector[C, Dilate[C, 13 - o, A]]], Translate[C, Vector[C, Dilate[C, 13 - o, A]]]], Polygon[Translate[W, Vector[C, A]], Translate[R', Vector[C, A]], Translate[C, Vector[C, A]]]) → t7
59. If(14 < o ≤ 15, Polygon[Translate[W', Vector[B, Dilate[B, 15 - o, L]]], Translate[V, Vector[B, Dilate[B, 15 - o, L]]], Translate[B, Vector[B, Dilate[B, 15 - o, L]]]], Polygon[Translate[W', Vector[B, L]], Translate[V, Vector[B, L]], Translate[B, Vector[B, L]]]) → t8
60. If(16 < o ≤ 17, Polygon[Translate[W'', Vector[H, Dilate[H, 17 - o, M]]], Translate[H, Vector[H, Dilate[H, 17 - o, M]]], Translate[U, Vector[H, Dilate[H, 17 - o, M]]]], Polygon[Translate[W'', Vector[H, M]], Translate[H, Vector[H, M]], Translate[U, Vector[H, M]]]) → t9
61. If(18 < o ≤ 19, Polygon[Translate[I, Vector[I, Dilate[I, 19 - o, N]]], Translate[W''', Vector[I, Dilate[I, 19 - o, N]]], Translate[T, Vector[I, Dilate[I, 19 - o, N]]]], Polygon[Translate[I, Vector[I, N]], Translate[W''', Vector[I, N]], Translate[T, Vector[I, N]]]) → t10
62. If(20 < o ≤ 21, Polygon[Translate[J, Vector[J, Dilate[J, 21 - o, O]]], Translate[C_1, Vector[J, Dilate[J, 21 - o, O]]], Translate[S, Vector[J, Dilate[J, 21 - o, O]]]], Polygon[Translate[J, Vector[J, O]], Translate[C_1, Vector[J, O]], Translate[S, Vector[J, O]]]) → t11
63. If(22 < o ≤ 23, Polygon[Translate[K, Vector[K, Dilate[K, 23 - o, B]]], Translate[F_1, Vector[K, Dilate[K, 23 - o, B]]], Translate[R, Vector[K, Dilate[K, 23 - o, B]]]], Polygon[Translate[K, Vector[K, B]], Translate[F_1, Vector[K, B]], Translate[R, Vector[K, B]]]) → t12
64. Centroid(poly3) → G_1
65. Centroid(poly1) → H_1
66. If(24 < o ≤ 25, Translate[poly1, Vector[H_1, Dilate[H_1, 25 - o, G_1]]], Translate[poly1, Vector[H_1, G_1]]) → poly4
67. Polygon(C, A, G, F, E, D) → poly5, c_3, a_2, g_2, f_2, e_2, d_2
68. Angle(A, C, B) → β