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勾股定理的无字证明

勾股定理的无字证明
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2019-06-01 19:34
圆多边形函数几何作图
所属专题 Proofs Without Words Proofs Without Words
①本范例通过动态演示半圆与直角三角形的关系，直观展示勾股定理的几何意义；②涉及的核心数学概念包括勾股定理、圆、多边形、中点、镜像、线段等；③作图要点运用了Point, Semicircle, Polygon, Midpoint, Mirror, Line, Centroid, Intersect, Rotate, Angle, If等GeoGebra命令与对象；④适合初中及以上学生及数学爱好者用于理解勾股定理的几何证明方法。

GeoGebra 命令及参数

  1. Point(yAxis) → A
  2. Semicircle(A, B) → c
  3. Point(c) → C
  4. Polygon(A, C, 4) → poly1, f, g, h, i, D, E
  5. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  6. Polygon(B, A, 4) → poly3, n, p, q, r, H, I
  7. Midpoint(j) → J
  8. Midpoint(m) → K
  9. Midpoint(l) → L
  10. Midpoint(k) → M
  11. Mirror(A, j) → A'
  12. Line(J, L) → s
  13. Line(A', B) → t
  14. Centroid(poly2) → N
  15. Intersect(s, t) → O
  16. Rotate(O, 90°, N) → O'
  17. Rotate(O', 90°, N) → O''
  18. Rotate(O'', 90°, N) → O'''
  19. Polygon(N, O, B, M) → q1, n_1, o_1, b, m_1
  20. Polygon(N, O', F, L) → q2, n_2, o', f_1, l_1
  21. Polygon(N, O'', G, K) → q3, n_3, o'', g_1, k_1
  22. Polygon(N, O''', C, J) → q4, n_4, o''', c_1, j_1
  23. Angle(C, B, A) → a
  24. If(0 < o ≤ 1, Rotate[Polygon[Translate[N, Vector[B, Dilate[B, 1 - o, I]]], Translate[O, Vector[B, Dilate[B, 1 - o, I]]], Translate[B, Vector[B, Dilate[B, 1 - o, I]]], Translate[M, Vector[B, Dilate[B, 1 - o, I]]]], (((o - 0) * 2) * a), Dilate[B, 1 - o, I]], Rotate[Polygon[Translate[N, I - B], Translate[O, I - B], Translate[B, I - B], Translate[M, I - B]], (2 * a), I]) → q5
  25. If(2 < o ≤ 3, Rotate[Polygon[Translate[N, Vector[F, Dilate[F, 3 - o, B]]], Translate[O', Vector[F, Dilate[F, 3 - o, B]]], Translate[F, Vector[F, Dilate[F, 3 - o, B]]], Translate[L, Vector[F, Dilate[F, 3 - o, B]]]], (((o - 2) * 2) * a), Dilate[F, 3 - o, B]], Rotate[Polygon[Translate[N, B - F], Translate[O', B - F], Translate[F, B - F], Translate[L, B - F]], (2 * a), B]) → q6
  26. If(4 < o ≤ 5, Rotate[Polygon[Translate[N, Vector[G, Dilate[G, 5 - o, A]]], Translate[O'', Vector[G, Dilate[G, 5 - o, A]]], Translate[G, Vector[G, Dilate[G, 5 - o, A]]], Translate[K, Vector[G, Dilate[G, 5 - o, A]]]], (((o - 4) * 2) * a), Dilate[G, 5 - o, A]], Rotate[Polygon[Translate[N, A - G], Translate[O'', A - G], Translate[G, A - G], Translate[K, A - G]], (2 * a), A]) → q7
  27. If(6 < o ≤ 7, Rotate[Polygon[Translate[N, Vector[C, Dilate[C, 7 - o, H]]], Translate[O''', Vector[C, Dilate[C, 7 - o, H]]], Translate[C, Vector[C, Dilate[C, 7 - o, H]]], Translate[J, Vector[C, Dilate[C, 7 - o, H]]]], (((o - 6) * 2) * a), Dilate[C, 7 - o, H]], Rotate[Polygon[Translate[N, H - C], Translate[O''', H - C], Translate[C, H - C], Translate[J, H - C]], (2 * a), H]) → q8
  28. Polygon(O, J, B) → t1, b_1, o_2, j_2
  29. Polygon(O', M, F) → t2, f_2, d, m_2
  30. Polygon(O'', L, G) → t3, g_2, e, l_2
  31. Polygon(O''', K, C) → t4, c_2, h_1, k_2
  32. Line(I, B) → i_1
  33. Intersect(i_1, l) → P
  34. Angle(P, B, O) → aa
  35. If(8 < o ≤ 9, Rotate[Polygon[Translate[O, Vector[B, Dilate[B, 9 - o, I]]], Translate[J, Vector[B, Dilate[B, 9 - o, I]]], Translate[B, Vector[B, Dilate[B, 9 - o, I]]]], ((-(o - 8)) * aa), Dilate[B, 9 - o, I]], Rotate[Polygon[Translate[O, I - B], Translate[J, I - B], Translate[B, I - B]], (-aa), I]) → t5
  36. If(10 < o ≤ 11, Rotate[Polygon[Translate[O', Vector[F, Dilate[F, 11 - o, B]]], Translate[M, Vector[F, Dilate[F, 11 - o, B]]], Translate[F, Vector[F, Dilate[F, 11 - o, B]]]], ((-(o - 10)) * aa), Dilate[F, 11 - o, B]], Rotate[Polygon[Translate[O', B - F], Translate[M, B - F], Translate[F, B - F]], (-aa), B]) → t6
  37. If(12 < o ≤ 13, Rotate[Polygon[Translate[O'', Vector[G, Dilate[G, 13 - o, A]]], Translate[L, Vector[G, Dilate[G, 13 - o, A]]], Translate[G, Vector[G, Dilate[G, 13 - o, A]]]], ((-(o - 12)) * aa), Dilate[G, 13 - o, A]], Rotate[Polygon[Translate[O'', A - G], Translate[L, A - G], Translate[G, A - G]], (-aa), A]) → t7
  38. If(14 < o ≤ 15, Rotate[Polygon[Translate[O''', Vector[C, Dilate[C, 15 - o, H]]], Translate[K, Vector[C, Dilate[C, 15 - o, H]]], Translate[C, Vector[C, Dilate[C, 15 - o, H]]]], ((-(o - 14)) * aa), Dilate[C, 15 - o, H]], Rotate[Polygon[Translate[O''', H - C], Translate[K, H - C], Translate[C, H - C]], (-aa), H]) → t8
  39. Centroid(poly3) → Q
  40. Centroid(poly1) → R
  41. Polygon(A, E, D, C) → q9, a_1, e_1, d_1, c_3
  42. If(16 < o ≤ 17, Rotate[Polygon[Translate[A, Vector[R, Dilate[R, 17 - o, Q]]], Translate[E, Vector[R, Dilate[R, 17 - o, Q]]], Translate[D, Vector[R, Dilate[R, 17 - o, Q]]], Translate[C, Vector[R, Dilate[R, 17 - o, Q]]]], (((o - 16) * 2) * a), Dilate[R, 17 - o, Q]], Rotate[Polygon[Translate[A, Q - R], Translate[E, Q - R], Translate[D, Q - R], Translate[C, Q - R]], (2 * a), Q]) → q10

AI 绘图提示语参考

在坐标轴上取一点作为半圆的端点，以该点和另一点为直径作半圆，在半圆上取一点，分别以该点与直径两端点构造三个正方形，并连接各边中点形成辅助线。将半圆内的三角形通过旋转和平移，使四个小正方形围绕中心旋转，动态展示勾股定理的几何证明。图形包含半圆、三个正方形、多个旋转后的多边形及辅助线，整体呈现对称的几何变换过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。