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无字证明：勾股定理

无字证明：勾股定理
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2019-06-09 08:48
圆多边形角几何作图
所属专题 Proofs Without Words Proofs Without Words
① 此 GeoGebra 范例无字演示勾股定理，通过正方形、半圆、多边形及交点构建几何图形，直观展示面积关系。② 核心概念包括勾股定理、平移、多边形面积、交点坐标。③ 作图要点涉及 Point、Semicircle、Segment、Polygon、Angle、Line、Intersect、OrthogonalLine、Circle 及 If 条件命令。④ 适合初中至高中几何教学，帮助学生视觉化理解定理证明过程。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Semicircle(A, B) → c
  4. Point(c) → C
  5. Segment(A, C) → f
  6. Segment(C, B) → g
  7. Segment(B, A) → h
  8. Polygon(A, C, 4) → poly1, i, j, k, l, G, H
  9. Polygon(C, B, 4) → poly2, m, n, p, q, I, J
  10. Polygon(B, A, 4) → poly3, r, s, t, a, K, L
  11. Angle(A, C, B) → α
  12. Line(K, A) → b
  13. Line(L, B) → d
  14. Line(C, h) → e
  15. Intersect(d, p) → F
  16. Intersect(d, e) → M
  17. Intersect(b, e) → N
  18. Intersect(b, j) → D
  19. Intersect(e, k) → E
  20. Polygon(A, N, C) → t2, c_1, a_2, n_1
  21. Polygon(D, N, C) → t3, c_2, d_2, n_2
  22. Polygon(C, M, B) → t4, b_1, c_3, m_1
  23. Polygon(C, J, F, M) → q1, c_4, j_1, f_2, m_2
  24. Polygon(B, F, I) → t5, i_1, b_2, f_3
  25. OrthogonalLine(C, h) → k_1
  26. Intersect(k_1, t) → P
  27. Intersect(k_1, h) → Q
  28. Circle(H, Segment[G, D]) → d_1
  29. Intersect(d_1, l, 1) → O
  30. OrthogonalLine(O, h) → f_1
  31. Intersect(f_1, k) → R
  32. Polygon(O, R, G, D, A) → poly4, o, r_1, g_1, d_3, a_1
  33. Polygon(O, R, H) → t1, h_1, o_1, r_2
  34. If(0 < u ≤ 1, Polygon[Translate[O, Dilate[A, 1 - u, P] - A], Translate[R, Dilate[A, 1 - u, P] - A], Translate[G, Dilate[A, 1 - u, P] - A], Translate[D, Dilate[A, 1 - u, P] - A], Translate[A, Dilate[A, 1 - u, P] - A]], Polygon[Translate[O, P - A], Translate[R, P - A], Translate[G, P - A], Translate[D, P - A], Translate[A, P - A]]) → poly5
  35. If(2 < u ≤ 3, Polygon[Translate[A, A - Dilate[A, 3 - u, N]], Translate[N, A - Dilate[A, 3 - u, N]], Translate[C, A - Dilate[A, 3 - u, N]]], Polygon[Translate[A, A - N], Translate[N, A - N], Translate[C, A - N]]) → t6
  36. If(4 < u ≤ 5, Polygon[Translate[O, Q - Dilate[Q, 5 - u, R]], Translate[R, Q - Dilate[Q, 5 - u, R]], Translate[H, Q - Dilate[Q, 5 - u, R]]], Polygon[Translate[O, Q - R], Translate[R, Q - R], Translate[H, Q - R]]) → t7
  37. If(6 < u ≤ 7, Polygon[Translate[D, K - Dilate[K, 7 - u, N]], Translate[N, K - Dilate[K, 7 - u, N]], Translate[C, K - Dilate[K, 7 - u, N]]], Polygon[Translate[D, K - N], Translate[N, K - N], Translate[C, K - N]]) → t8
  38. If(8 < u ≤ 9, Polygon[Translate[B, P - Dilate[P, 9 - u, B]], Translate[F, P - Dilate[P, 9 - u, B]], Translate[I, P - Dilate[P, 9 - u, B]]], Polygon[Translate[B, P - B], Translate[F, P - B], Translate[I, P - B]]) → t9
  39. If(10 < u ≤ 11, Polygon[Translate[C, B - Dilate[B, 11 - u, M]], Translate[M, B - Dilate[B, 11 - u, M]], Translate[B, B - Dilate[B, 11 - u, M]]], Polygon[Translate[C, B - M], Translate[M, B - M], Translate[B, B - M]]) → t10
  40. If(12 < u ≤ 13, Polygon[Translate[C, L - Dilate[L, 13 - u, M]], Translate[J, L - Dilate[L, 13 - u, M]], Translate[F, L - Dilate[L, 13 - u, M]], Translate[M, L - Dilate[L, 13 - u, M]]], Polygon[Translate[C, L - M], Translate[J, L - M], Translate[F, L - M], Translate[M, L - M]]) → q2
  41. Segment(F, B) → l_1
  42. Segment(M, N) → p_1
  43. Segment(D, A) → q_1
  44. Segment(R, O) → s_1

AI 绘图提示语参考

请绘制一个直角三角形，并展示其三条边上的正方形。图形中应包含半圆和线段，用于表示面积关系。通过动态的几何构造，如多边形、交点、角和正交线，展示勾股定理的证明过程。帮助学生直观理解a² + b² = c²的面积关系，使定理的证明可视化。
以上提示语为根据原始绘图命令整理而来，仅供参考。