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毕达哥拉斯定理的无字几何证明：圆与相似三角形拼合

毕达哥拉斯定理的无字几何证明：圆与相似三角形拼合
几何画板及绘图内容加载中，请稍等……
2021-03-01 07:22
测试改名0圆多边形角几何作图
该范例通过一幅“无字证明”式的几何拼合图形直观展示直角三角形中面积关系，从而呈现毕达哥拉斯定理的结论。核心概念包括：直角三角形、相似三角形、圆与角关系、以及利用作图构造出的中点、重心与镜面对称来形成可对比的面积/长度结构。作图要点从构造摘要可见：使用 Intersect 获取关键交点，Point/Line/Circle 确定几何骨架；用 Polygon 构成多边形区域以便比较与拼合；借助 Midpoint、Centroid 构造辅助点；采用 Mirror 与 Rotate 进行对称与旋转拼接；用 OrthogonalLine/LineBisector/Angle 明确直角与角平分等约束；并通过 Circle 与相应交点生成与圆相关的相似/角度配置。适合中学高年级到竞赛训练使用，也可用于几何直观教学、定理可视化与课堂探究活动，帮助学生用“图形不等式式”思维理解定理。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 5) → poly1, f, g, h, i, j, C, D, E
  4. Centroid(poly1) → F
  5. Midpoint(j) → G
  6. Midpoint(i) → H
  7. Midpoint(h) → I
  8. Midpoint(g) → J
  9. Centroid(poly1) → K
  10. Midpoint(f) → L
  11. Point(Segment[K, L]) → M
  12. Polygon(A, G, F, M) → q1, a, g_1, f_1, m
  13. Polygon(L, A, M) → t1, m_1, l, a_1
  14. Mirror(M, m_1) → M'
  15. Line(A, M') → k
  16. OrthogonalLine(B, xAxis) → n
  17. Intersect(k, n) → N
  18. Polygon(N, A, 5) → poly2, p, q, r, s, t, O, P, Q
  19. Centroid(poly2) → R
  20. Circle(F, Segment[R, N]) → c
  21. Line(E, F) → b
  22. Line(F, D) → d
  23. Line(F, C) → e
  24. Line(F, B) → h_1
  25. Line(F, A) → i_1
  26. Intersect(c, i_1, 2) → S
  27. Intersect(c, b, 1) → T
  28. Intersect(c, d, 2) → U
  29. Intersect(c, e, 2) → V
  30. Intersect(c, h_1, 2) → W
  31. Rotate(M, (2 * π / 5), K) → M'_1
  32. Polygon(L, F, M'_1, B) → q2, l_1, f_2, m'_1, b_1
  33. Line(W, V) → j_1
  34. Intersect(b, j_1) → Z
  35. LineBisector(M'_1, Z) → k_1
  36. LineBisector(B, W) → n_1
  37. Intersect(k_1, n_1) → A_1
  38. Angle(W, A_1, B) → α
  39. If(0 ≤ o ≤ 1, Polygon[Rotate[L, ((-o) * α), A_1], Rotate[F, ((-o) * α), A_1], Rotate[M'_1, ((-o) * α), A_1], Rotate[B, ((-o) * α), A_1]]) → q3
  40. If(0 ≤ o ≤ 1, Rotate[L, ((-o) * α), A_1]) → B_1
  41. Rotate(If[0 ≤ o ≤ 1, Polygon[Rotate[L, ((-o) * α), A_1], Rotate[A, ((-o) * α), A_1], Rotate[M, ((-o) * α), A_1]]], (o * π), B_1) → t2
  42. Rotate(If[0 ≤ o ≤ 1, Rotate[M, ((-o) * α), A_1]], (o * π), B_1) → C_1
  43. Rotate(Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[A, ((-o) * α), A_1], Rotate[G, ((-o) * α), A_1], Rotate[F, ((-o) * α), A_1], Rotate[M, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1) → q4
  44. Rotate(Rotate[If[0 ≤ o ≤ 1, Rotate[G, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1) → D_1
  45. Rotate(M, (-2 * π / 5), K) → M'_2
  46. Polygon(E, G, M'_2) → t3, m'_2, e_1, g_2
  47. Rotate(Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[E, ((-o) * α), A_1], Rotate[G, ((-o) * α), A_1], Rotate[M'_2, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1) → t4
  48. Rotate(Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[M'_2, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1) → E_1
  49. Polygon(H, E, M'_2, F) → q5, h_2, e_2, p_1, f_3
  50. Rotate(Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[H, ((-o) * α), A_1], Rotate[E, ((-o) * α), A_1], Rotate[M'_2, ((-o) * α), A_1], Rotate[F, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1) → q6
  51. Rotate(M, (6 * π / 5), K) → M'_3
  52. Rotate(Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[H, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1) → F_1
  53. Polygon(D, H, M'_3) → t5, m'_3, d_1, h_3
  54. Rotate(Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[D, ((-o) * α), A_1], Rotate[H, ((-o) * α), A_1], Rotate[M'_3, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1) → t6
  55. Rotate(Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[M'_3, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1) → G_1
  56. Polygon(I, F, M'_3, D) → q7, i_2, f_4, q_1, d_2
  57. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[I, ((-o) * α), A_1], Rotate[F, ((-o) * α), A_1], Rotate[M'_3, ((-o) * α), A_1], Rotate[D, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1) → q8
  58. Rotate(M, (4 * π / 5), K) → M'_4
  59. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[I, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1) → H_1
  60. Polygon(M'_4, I, C) → t7, c_1, m'_4, i_3
  61. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[M'_4, ((-o) * α), A_1], Rotate[I, ((-o) * α), A_1], Rotate[C, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1], (o * π), H_1) → t8
  62. Polygon(F, M'_4, C, J) → q9, f_5, r_1, c_2, j_2
  63. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[M'_4, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1], (o * π), H_1) → I_1
  64. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[F, ((-o) * α), A_1], Rotate[M'_4, ((-o) * α), A_1], Rotate[C, ((-o) * α), A_1], Rotate[J, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1], (o * π), H_1], ((-o) * π), I_1) → q10
  65. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Rotate[J, ((-o) * α), A_1]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1], (o * π), H_1], ((-o) * π), I_1) → J_1
  66. Polygon(B, M'_1, J) → t9, j_3, b_2, s_1
  67. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[If[0 ≤ o ≤ 1, Polygon[Rotate[B, ((-o) * α), A_1], Rotate[M'_1, ((-o) * α), A_1], Rotate[J, ((-o) * α), A_1]]], (o * π), B_1], ((-o) * π), C_1], (o * π), D_1], ((-o) * π), E_1], (o * π), F_1], ((-o) * π), G_1], (o * π), H_1], ((-o) * π), I_1], (o * π), J_1) → t10

AI 绘图提示语参考

绘制一个以原点为直角顶点的直角三角形，并构造一个正五边形作为基础框架。在五边形内部，通过连接各边中点与重心，划分出多个小三角形和四边形区域。以重心为圆心，过某顶点作圆，圆与五边形各顶点连线相交，形成五个交点。利用这些交点构造角平分线，确定旋转中心，并围绕该中心将部分多边形旋转复制，形成对称的拼图效果。整个图形通过旋转和镜像操作，将不同区域重新排列，直观展示面积关系，体现勾股定理的几何证明。
以上提示语为根据原始绘图命令整理而来，仅供参考。