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勾股定理剖分证明：平移变换演示

勾股定理剖分证明：平移变换演示
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2015-07-27 09:27
测试改名0多边形向量变换几何作图
该范例以动态几何方式演示勾股定理的剖分证明，通过将一个直角三角形及其相关多边形进行平移、旋转与对称变换，直观展示两直角边平方和等于斜边平方的面积守恒关系。核心概念包括勾股定理、三角形面积、多边形剖分、向量平移、对称与旋转、几何变换下的不变量。作图时先用 Segment、Point、Angle、Polygon 构造直角三角形与边长对象 a、b、c，并用 Midpoint、Vector、Translate、Dilate、Circle、Semicircle、Ray、Intersect、Line 生成剖分块、辅助圆、半圆、射线和交点，形成 poly2、poly3 等可移动多边形及 D'、C'、E'、B' 等平移像。适合初中至高中几何教学、数学实验课、GeoGebra 课件制作与定理证明可视化。

GeoGebra 命令及参数

  1. Segment(A, B) → c
  2. Semicircle(A, B) → c_2
  3. Point(c_2) → C
  4. Segment(B, C) → b
  5. Segment(C, A) → a
  6. Angle(A, C, B) → α
  7. Polygon(C, B, 4) → small_1, e, f, g, h, D, E
  8. Polygon(A, C, 4) → poly2, i, j, k, l, F, G
  9. Polygon(B, A, 4) → poly3, m, n, p, q, H, I
  10. Midpoint(D, C) → J
  11. Midpoint(F, A) → L
  12. Midpoint(I, A) → K
  13. Vector(J, K) → v
  14. Translate(D, v) → D'
  15. Translate(C, v) → C'
  16. Translate(E, v) → E'
  17. Translate(B, v) → B'
  18. Ray(B', D') → r
  19. Ray(D', E') → s
  20. Ray(E', C') → t
  21. Ray(C', B') → a_1
  22. Segment(I, H) → b_1
  23. Segment(H, A) → c_1
  24. Segment(B, I) → d_1
  25. Intersect(a_1, b_1) → M
  26. Intersect(t, c_1) → N
  27. Intersect(r, d_1) → O
  28. Intersect(c, s) → P
  29. Dilate(D', 1 - equals / 100, A) → D''
  30. Circle(G, Segment[B', M]) → e_1
  31. Segment(F, G) → g_1
  32. Segment(A, G) → h_1
  33. Segment(C, F) → i_1
  34. Intersect(e_1, h_1, 1) → Q
  35. Circle(A, Segment[G, Q]) → k_1
  36. Circle(C, Segment[G, Q]) → q_1
  37. Circle(F, Segment[G, Q]) → r_1
  38. Intersect(e_1, g_1, 1) → R
  39. Intersect(r_1, g_1, 1) → S
  40. Intersect(q_1, a, 1) → T
  41. Intersect(k_1, a, 1) → U
  42. Intersect(k_1, h_1, 1) → V
  43. Intersect(q_1, i_1, 1) → W
  44. Intersect(r_1, i_1, 1) → Z
  45. Polygon(F, W, L, S) → yellow_1, y1A, y1D, y1B, y1C
  46. Polygon(S, L, Q, G) → green_1, g1B, g1C, g1D, g1A
  47. Polygon(L, U, A, Q) → red_1, r1B, r1C, r1D, r1A
  48. Polygon(U, L, W, C) → blue_1, b1D, b1A, b1B, b1C
  49. Dilate(D', equals / 100, A) → D''_1
  50. Vector(A, D''_1) → v_1
  51. Translate(L, v_1) → L'
  52. Translate(U, v_1) → U'
  53. Translate(A, v_1) → A'
  54. Translate(Q, v_1) → Q'
  55. Polygon(L', U', A', Q') → red_1', r1B', r1C', r1D', r1A'
  56. Dilate(P, equals / 100, W) → P'
  57. Vector(W, P') → j_1
  58. Translate(U, j_1) → U'_1
  59. Translate(L, j_1) → L'_1
  60. Translate(W, j_1) → W'
  61. Translate(C, j_1) → C'_1
  62. Polygon(U'_1, L'_1, W', C'_1) → blue_1', b1D', b1A', b1B', b1C'
  63. Dilate(K, equals / 100, J) → K'
  64. Vector(J, K') → m_1
  65. Translate(small_1, m_1) → small_1'
  66. Dilate(H, equals / 100, L) → H'
  67. Vector(L, H') → n_1
  68. Translate(F, n_1) → F'
  69. Translate(W, n_1) → W'_1
  70. Translate(L, n_1) → L'_2
  71. Translate(S, n_1) → S'
  72. Polygon(F', W'_1, L'_2, S') → yellow_1', y1A', y1D_1, y1B_1, y1C_1
  73. Dilate(B', equals / 100, G) → B''
  74. Vector(G, B'') → w
  75. Translate(S, w) → S'_1
  76. Translate(L, w) → L'_3
  77. Translate(Q, w) → Q'_1
  78. Translate(G, w) → G'
  79. Polygon(S'_1, L'_3, Q'_1, G') → green_1', g1B', g1C', g1D', g1A'
  80. Dilate(K, equals / 100, L) → K'_1
  81. Vector(L, K'_1) → u
  82. Translate(poly2, u) → poly2'
  83. Midpoint(A, H) → A_1
  84. Midpoint(I, H) → B_1
  85. Midpoint(B, I) → C_1
  86. Line(C_1, b) → d
  87. Line(P, a) → f_1
  88. Line(A_1, d) → l_1
  89. Line(B_1, f_1) → p_1
  90. Intersect(d, f_1) → D_1
  91. Intersect(l_1, p_1) → E_1
  92. Intersect(f_1, l_1) → F_1
  93. Intersect(d, p_1) → G_1
  94. Intersect(l_1, b_1) → H_1
  95. Circle(E, Segment[C_1, G_1]) → s_1
  96. Circle(D, Segment[C_1, G_1]) → t_1
  97. Circle(B, Segment[C_1, G_1]) → d_2
  98. Circle(C, Segment[C_1, G_1]) → e_2
  99. Intersect(b, e) → I_1
  100. Intersect(b, e) → J_1
  101. Intersect(b, e) → K_1
  102. Intersect(d_2, e, 1) → L_1
  103. Intersect(d_2, f, 1) → M_1
  104. Intersect(e_2, e, 1) → N_1
  105. Intersect(e_2, h, 1) → O_1
  106. Intersect(s_1, h, 1) → P_1
  107. Intersect(s_1, g, 1) → Q_1
  108. Intersect(t_1, g, 1) → R_1
  109. Intersect(t_1, f, 1) → S_1
  110. Polygon(D, S_1, J, Q_1) → yellow_2, y2B, y2C, y2D, y2A
  111. Polygon(S_1, J, L_1, B) → blue_2, b2A, b2D, b2C, b2B
  112. Polygon(L_1, J, O_1, C) → red_2, r2B, r2A, r2D, r2C
  113. Polygon(Q_1, E, O_1, J) → green_2, g2A, g2D, g2C, g2B
  114. Dilate(A, equals / 100, J) → A'_1
  115. Vector(J, A'_1) → u_1
  116. Dilate(B, equals / 100, J) → B'_1
  117. Vector(J, B'_1) → w_1
  118. Dilate(H, equals / 100, J) → H'_1
  119. Vector(J, H'_1) → u_2
  120. Dilate(I, equals / 100, J) → I'
  121. Vector(J, I') → v_2
  122. Translate(Q_1, v_2) → T_1
  123. Translate(E, v_2) → E'_1
  124. Translate(O_1, v_2) → U_1
  125. Translate(J, v_2) → J'
  126. Polygon(T_1, E'_1, U_1, J') → green_2', g2A', g2D', g2C', g2B'
  127. Translate(S_1, u_1) → V_1
  128. Translate(J, u_1) → J'_1
  129. Translate(L_1, u_1) → W_1
  130. Translate(B, u_1) → B'_2
  131. Polygon(V_1, J'_1, W_1, B'_2) → blue_2', b2A', b2D', b2C', b2B'
  132. Translate(L_1, w_1) → Z_1
  133. Translate(J, w_1) → J'_2
  134. Translate(O_1, w_1) → A_2
  135. Translate(C, w_1) → C'_2
  136. Polygon(Z_1, J'_2, A_2, C'_2) → red_2', z_1, r2A', r2D', r2C'
  137. Translate(D, u_2) → D'_1
  138. Translate(S_1, u_2) → B_2
  139. Translate(J, u_2) → J'_3
  140. Translate(Q_1, u_2) → C_2
  141. Polygon(D'_1, B_2, J'_3, C_2) → yellow_2', y2B', y2C', y2D', y2A'

AI 绘图提示语参考

绘制一个直角三角形ABC，其中AB为斜边，AC和BC为直角边。通过平移、旋转和反射，将三角形及其相关多边形分解并重新组合，展示直角三角形的两个直角边的平方和等于斜边的平方。具体步骤包括构造直角三角形、创建多边形、平移向量、旋转和反射点，最终形成四个颜色不同的多边形，分别对应原三角形的各部分。
以上提示语为根据原始绘图命令整理而来，仅供参考。