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勾股定理剪切法证明

勾股定理剪切法证明
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2019-08-16 05:20
测试改名0圆函数几何作图滑块动画
①本范例通过剪切法演示勾股定理的证明过程；②涉及核心数学概念包括勾股定理、二次函数、等腰直角三角形、几何平均数、反射、圆；③构造要点：使用Intersect, Point, Semicircle, Midpoint, Circle, Rotate, Line, Vector, Length, If, Polygon, Angle等GeoGebra命令和对象；④适合高中数学教学和几何证明学习。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Semicircle(A, B) → h
  4. Midpoint(A, B) → O
  5. Circle(O, B) → o
  6. Point(h) → C
  7. Rotate(C, 90*°, A) → D
  8. Rotate(A, 270*°, C) → E
  9. Line(D, E) → m
  10. Vector(D, E) → v_b
  11. Length(v_b) → k_b
  12. Rotate(C, 270*°, B) → F
  13. Rotate(B, 90*°, C) → G
  14. Line(F, G) → n
  15. Intersect(m, n) → H
  16. Line(H, yAxis) → hc
  17. Vector(E, H) → u_a
  18. Length(u_a) → ku_b
  19. Vector(G, H) → u_b
  20. Length(u_b) → ku_a
  21. Vector(F, G) → v_a
  22. Length(v_a) → k_a
  23. Rotate(B, 270*°, A) → I
  24. Vector(A, I) → v_c
  25. Length(v_c) → k_c
  26. Circle(H, (ku_a * ku_b / k_c)) → d
  27. Intersect(d, hc) → K, L
  28. Vector(H, L) → v_h
  29. Rotate(A, 90*°, B) → J
  30. If(a < ka_1, Point[F, Vector[(a / ka_1 * u_b)]], If[a < ka_2, Point[F, u_b], Point[F, Vector[u_b + ((a - ka_2) / (1 - ka_2) * v_c)]]]) → F'
  31. Polygon(B, C, G, F) → P, b_1, c, g, f
  32. If(a < ka_1, Point[G, Vector[(a / ka_1 * u_b)]], If[a < ka_2, Point[H, Vector[((a - ka_1) / (ka_2 - ka_1) * v_h)]], Point[L, Vector[((a - ka_2) / (1 - ka_2) * v_c)]]]) → G'
  33. If(a < ka_1, C, If[a < ka_2, Point[C, Vector[((a - ka_1) / (ka_2 - ka_1) * v_h)]], Point[C, Vector[v_h + ((a - ka_2) / (1 - ka_2) * v_c)]]]) → Ca'
  34. Vector(G', Ca') → u
  35. If(a < ka_1, B, If[a < ka_2, B, Point[B, Vector[((a - ka_2) / (1 - ka_2) * v_c)]]]) → B'
  36. Vector(F', B') → v
  37. Polygon(B', Ca', G', F') → Q, b', c', g', f'
  38. If(b < kb_1, Point[E, Vector[(b / kb_1 * u_a)]], If[b < kb_2, Point[H, Vector[((b - kb_1) / (kb_2 - kb_1) * v_h)]], Point[L, Vector[((b - kb_2) / (1 - kb_2) * v_c)]]]) → E'
  39. If(b < kb_1, Point[D, Vector[(b / kb_1 * u_a)]], If[b < kb_2, Point[D, u_a], Point[D, Vector[u_a + ((b - kb_2) / (1 - kb_2) * v_c)]]]) → D'
  40. If(b < kb_1, C, If[b < kb_2, Point[C, Vector[((b - kb_1) / (kb_2 - kb_1) * v_h)]], Point[C, Vector[v_h + ((b - kb_2) / (1 - kb_2) * v_c)]]]) → Cb'
  41. If(b < kb_1, A, If[b < kb_2, A, Point[A, Vector[((b - kb_2) / (1 - kb_2) * v_c)]]]) → A'
  42. Polygon(A, C, E, D) → R, a_1, c_1, e, d_1
  43. Polygon(A', Cb', E', D') → S, a', cb', e', d'
  44. Polygon(A, B, J, I) → T, a_2, b_2, j, i
  45. Angle(A, C, B) → α
  46. Midpoint(C, B) → M
  47. Midpoint(A, C) → N
  48. Midpoint(a_2) → U