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勾股定理的分割证明

勾股定理的分割证明
几何画板及绘图内容加载中，请稍等……
2021-11-19 12:13
测试改名0多边形向量三角函数几何作图
①本范例演示了通过分割图形证明勾股定理的几何方法；②涉及的核心数学概念包括勾股定理、图形分割、面积计算、向量与变换；③作图要点提炼自构造摘要，使用了Segment（线段）、Intersect（交点）、Polygon（多边形）、PointIn（点在图形内）、Point（点）、Vector（向量）、Line（直线）、Translate（平移）、Sequence（序列）等GeoGebra命令与对象；④适合高中数学教学，用于直观展示勾股定理的多种证明方法，加深学生对定理的理解和应用。

GeoGebra 命令及参数

  1. Segment((a, (-a)), (0, (-a))) → l
  2. Intersect(xAxis, yAxis) → O
  3. Segment(O, (0, (-a))) → k
  4. Segment(O, (a, 0)) → j
  5. Polygon({O, O + (a, 0), O + (a, (-a)), O + (0, (-a))}) → q1
  6. PointIn(Polygon[{O + (a / 2, 0), O + (a, 0), O + (a, ((-a)) / 2), O + (a / 2, ((-a)) / 2)}]) → C
  7. Point(Segment[O + (1.25, 0), O + (2.99, 0)]) → D
  8. Vector(O - (0, a), D) → u
  9. Line(C, C + u) → p
  10. Intersect(p, l) → H_{3}
  11. Vector(O + (a, 0) + (0, x(O - D)), O) → v
  12. Line(C - u, C - u + v) → j_8
  13. Intersect(j_8, l) → H_5
  14. Intersect(j_8, k) → H_4
  15. Line(C, C + v) → n
  16. Intersect(n, j) → F_2
  17. Polygon({O, D, D + (0, x(D - O)), O + (0, x(D - O))}) → q2
  18. Polygon({C, C - v, C - v + u, C + u}) → q3
  19. Translate(q2, Vector[(-u) - v]) → q2'
  20. Polygon({(0, 0), (b, 0), (b, a)}) → t1
  21. Segment((a, (-a)), (a, 0)) → m
  22. Intersect(p, j) → E_1
  23. Intersect(n, m) → G_1
  24. Intersect(n, k) → F_1
  25. Intersect(p, m) → E_2
  26. Intersect(n, l) → G_2
  27. Intersect(p, k) → H_2
  28. Polygon(I_1, I_1 + E_1 - C, I_1 + P - C, I_1 + G_1 - C) → q4, i_1, f, g, h
  29. Polygon(I_1, I_1 + E_2 - C, I_1 + G_1 - C) → t2, i, q, r
  30. Polygon(I_1, I_1 + E_2 - C, I_1 + Q - C, I_1 + G_2 - C) → q5, s, t, c, d
  31. Polygon(I_1, I_1 + E_1 - C, I_1 + P - C, I_1 + Q - C, I_1 + G_2 - C) → poly1, e, f_1, g_1, h_1, j_1
  32. Polygon(I_2, I_2 + E_1 - C, I_2 + O - C, I_2 + F_1 - C) → q6, i_2, k_1, l_1, m_1
  33. Polygon(I_2, I_2 + E_1 - C, I_2 + F_2 - C) → t3, n_1, p_1, q_1
  34. Polygon(I_2, I_2 + E_2 - C, I_2 + P - C, I_2 + F_2 - C) → q7, r_1, s_1, t_1, a_1
  35. Polygon(I_2, I_2 + E_2 - C, I_2 + P - C, I_2 + O - C, I_2 + F_1 - C) → poly2, b_1, c_1, d_1, e_1, f_2
  36. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + H_4 - C, I_3 + H_5 - C, I_3 + H_{3} - C) → poly3, g_2, h_2, j_2, k_2, i_3, k_8
  37. Polygon(I_3, I_3 + F_1 - C, I_3 + R - C, I_3 + H_{3} - C) → q8, l_2, m_2, n_2, p_2
  38. Polygon(I_3, I_3 + F_1 - C, I_3 + H_2 - C) → t4, q_2, r_2, s_2
  39. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + H_2 - C) → q9, t_2, a_2, b_2, c_2
  40. Polygon(I_4, I_4 + H_2 - C, I_4 + R - C, I_4 + Q - C, I_4 + G_1 - C) → poly4, i_4, d_2, e_2, f_3, g_3
  41. Polygon(I_4, I_4 + H_2 - C, I_4 + R - C, I_4 + G_2 - C) → q10, h_3, j_3, k_3, l_3
  42. Polygon(I_4, I_4 + H_{3} - C, I_4 + Q - C, I_4 + G_1 - C) → q11, m_3, n_3, p_3, q_3
  43. Polygon(I_4, I_4 + H_{3} - C, I_4 + G_2 - C) → t5, r_3, s_3, t_3
  44. Segment(O, O + (b, 0)) → a_3
  45. Segment(O + (b, b), O + (b, 0)) → b_3
  46. Segment(O + (b, b), O + (0, b)) → c_3
  47. Segment(O, O + (0, b)) → d_3
  48. Line(C + u, C + u + v) → e_3
  49. Line(C + v, C + u + v) → f_4
  50. Intersect(f_4, d_3) → K_1
  51. Intersect(f_4, c_3) → K_2
  52. Intersect(e_3, c_3) → K_3
  53. Intersect(e_3, b_3) → K_4
  54. Intersect(e_3, d_3) → K_{12}
  55. Intersect(f_4, j) → K_{10}
  56. Intersect(p, c_3) → K_{11}
  57. Intersect(n, d_3) → K_8
  58. Intersect(p, b_3) → K_5
  59. Intersect(n, a_3) → K_7
  60. Intersect(p, a_3) → K_6
  61. Intersect(n, b_3) → K_9
  62. Polygon(J_1, J_1 + E - O, J_1 + K_9 - O, J_1 + K_8 - O) → q12, g_4, h_4, j_4, k_4
  63. Polygon(J_1, J_1 + K_7 - O, J_1 + K_8 - O) → t6, l_4, m_4, n_4
  64. Polygon(H_1, H_1 + K_2 - F, H_1 + K_8 - F, H_1 + K_9 - F) → t7, p_4, q_4, r_4, s_4
  65. Polygon(L_1 + K_1 - G, L_1 + K_2 - G, L_1) → t8, t_4, a_4, b_4
  66. Polygon(H_1, H_1 + K_2 - F, H_1 + K_1 - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + E - F) → poly6, e_4, f_5, g_5, h_5, i_5, j_5
  67. Polygon(H_1, H_1 + K_2 - F, H_1 + K_1 - F, H_1 + O - F, H_1 + E - F) → poly7, k_5, l_5, m_5, n_5, p_5
  68. Polygon(H_1, H_1 + K_2 - F, H_1 + K_{10} - F, H_1 + E - F) → q13, c_4, d_4, q_5, r_5
  69. Polygon(L_1, L_1 + O - G, L_1 + K_{10} - G, L_1 + K_2 - G) → q14, s_5, t_5, a_5, b_5
  70. Sequence(Sequence[Translate[q1, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l1, {{9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}}
  71. Polygon(L_1, L_1 + F - G, L_1 + K_5 - G, L_1 + K_6 - G, L_1 + O - G) → poly5, c_5, d_5, e_5, f_6, g_6
  72. Polygon(M, M + K_5 - E, M + K_6 - E) → t9, h_6, m_6, i_6
  73. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + K_6 - F, H_1 + K_5 - F) → poly8, j_6, k_6, l_6, n_6, p_6, a_8
  74. Polygon(L_1, L_1 + K_3 - G, L_1 + K_4 - G, L_1 + E - G, L_1 + O - G) → poly9, q_6, r_6, s_6, t_6, a_6
  75. Polygon(H_1, H_1 + K_3 - F, H_1 + K_4 - F) → t10, b_6, c_6, d_6
  76. Polygon(H_1, H_1 + G - F, H_1 + K_{12} - F, H_1 + K_4 - F) → q15, e_6, f_7, g_7, h_7
  77. Polygon(J_1, J_1 + E - O, J_1 + K_4 - O, J_1 + K_{12} - O) → q16, i_7, j_7, k_7, l_7
  78. Polygon(L_1, L_1 + (b, 0), L_1 + (b, (-b)), L_1 + (0, (-b))) → q17, m_7, n_7, p_7, q_7
  79. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_9 - F) → q18, r_7, s_7, t_7, a_7
  80. Polygon(H_1, H_1 + K_{11} - F, H_1 + K_6 - F, H_1 + E - F) → q19, b_7, c_7, d_7, e_7
  81. Polygon(L_1, L_1 + K_{11} - G, L_1 + K_6 - G, L_1 + O - G) → q20, f_8, g_8, h_8, i_8
  82. Polygon({I_5}) → poly10
  83. Polygon(I_5, I_5 + H_4 - H, I_5 + H_5 - H) → t11, l_8, m_8, n_8
  84. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + R - C, I_3 + H_{3} - C) → poly11, p_8, q_8, r_8, s_8, t_8
  85. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + E - F) → poly12, b_8, c_8, d_8, e_8, f_9
  86. Polygon(J_1, J_1 + K_1 - O, J_1 + K_2 - O, J_1 + K_3 - O, J_1 + K_4 - O, J_1 + E - O) → poly13, g_9, h_9, i_9, j_9, k_9, l_9
  87. Sequence(Sequence[Translate[q2, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l2, {{4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}}
  88. Sequence(Sequence[Translate[q1, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l1', {{9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}}
  89. Polygon({O, B, B - (0, b), O - (0, b)}) → q21
  90. Polygon({A, B, B + (a, 0), A + (a, 0)}) → q22
  91. Polygon({O, A, Rotate[O, -90°, A], Rotate[A, 90°, O]}) → q23
  92. Sequence(Sequence[Translate[q3, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l3, {{13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}}

AI 绘图提示语参考

在直角坐标系中取原点为交点，画两条互相垂直的坐标轴。以原点为底角，取对称的上、下端点形成一个边长为 a 的正方形并画出其四边形。再取一段水平线上的点作为参考，连结以原点为起点的向量并通过该向量确定一条与前述边相交的直线，得到若干交点。以这些交点与原点、水平参考点等为顶点组合多个不相交或部分重叠的多边形片，表示勾勒毕达哥拉斯定理的几何切割证明。对其中一组区域做平移，使片段形成对应的重叠/拼合关系，最终得到由多块多边形构成的拼图式图形。
以上提示语为根据原始绘图命令整理而来，仅供参考。