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勾股定理的证明 by Sumesh S S

勾股定理的证明 by Sumesh S S
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2016-01-14 02:06
测试改名0解析几何矩阵几何作图
①本范例演示了勾股定理的几何证明方法，通过构造直角三角形及其相关图形来验证定理；②涉及的核心数学概念包括直角三角形、面积、坐标、矩阵等；③作图要点——从下方『构造摘要』提炼用到了Polygon（多边形）、Intersect（交点）、Segment（线段）、Line（直线）、Translate（平移）、If（条件判断）等GeoGebra命令/对象；④适合高中阶段数学学习或数学爱好者用于深入理解勾股定理及其应用。

GeoGebra 命令及参数

  1. Polygon(C, A, B) → poly1, b, c, a
  2. Intersect(c, b) → F
  3. Intersect(a, b) → G
  4. Intersect(a, c) → J
  5. Polygon(G, J, 4) → poly3, h, i, j, k, K, L
  6. Polygon(F, G, 4) → poly2, d, e, f, g, D, E
  7. Segment(D, F) → l
  8. Segment(E, G) → m
  9. Intersect(l, m) → H
  10. Intersect(m, poly2) → I, M
  11. Line(H, a) → n
  12. Intersect(e, poly2) → N, O
  13. Intersect(n, e) → P
  14. Intersect(n, g) → Q
  15. Segment(P, Q) → p
  16. Line(H, i) → q
  17. Intersect(q, b) → R
  18. Intersect(q, poly2) → S, T
  19. Polygon(J, F, 4) → poly4, r, s, t_1, a_1, U, V
  20. Segment(S, R) → b_1
  21. Polygon(O, P, H, S) → poly5, o, p_1, h_1, s_1
  22. Polygon(P, G, R, H) → poly6, p_2, g_1, r_1, h_2
  23. Polygon(S, H, Q, I) → poly7, s_2, h_3, q_1, i_1
  24. Polygon(H, R, F, Q) → poly8, h_4, r_2, f_1, q_2
  25. Intersect(j, i) → W
  26. Intersect(j, k) → Z
  27. Intersect(b, d) → C_1
  28. Intersect(b_1, r_1) → D_1
  29. Intersect(b_1, r_1) → E_1
  30. Translate(G, (1, 0.75)) → G'
  31. Translate(R, (1, 0.75)) → R'
  32. Translate(G', (1, 0.75)) → G''
  33. Translate(R', (1, 0.75)) → R''
  34. Translate(G'', (0.24, 0.18)) → G'''
  35. Translate(R'', (0.24, 0.18)) → R'''
  36. Translate(H, (2.24, 1.68)) → H'''
  37. Translate(G''', (2.76, -3.68)) → H_1
  38. Translate(R''', (2.76, -3.68)) → I_1
  39. Translate(H''', (2.76, -3.68)) → J_1
  40. Translate(O, (2, 1.5)) → O'
  41. Translate(P, (2, 1.5)) → P'_1
  42. Translate(S, (2, 1.5)) → S'
  43. Translate(O, (4, 3)) → O'_1
  44. Translate(P, (4, 3)) → P'_2
  45. Translate(S, (4, 3)) → S'_1
  46. Translate(O, (6, 4.5)) → O'_2
  47. Translate(P, (6, 4.5)) → P'_3
  48. Translate(S, (6, 4.5)) → S'_2
  49. Translate(O, (6, 4.5)) → O'_3
  50. Translate(P, (6, 4.5)) → P'_4
  51. Translate(S, (6, 4.5)) → S'_3
  52. Translate(O, (2, 1.5)) → O'_4
  53. Translate(P, (2, 1.5)) → P'_5
  54. Translate(H, (2, 1.5)) → H'_3
  55. Translate(S, (2, 1.5)) → S'_4
  56. Translate(O, (4, 3)) → O'_5
  57. Translate(P, (4, 3)) → P'_6
  58. Translate(H, (4, 3)) → H'_1
  59. Translate(S, (4, 3)) → S'_5
  60. Translate(O, (6.24, 4.68)) → O'_6
  61. Translate(P, (6.24, 4.68)) → P'_7
  62. Translate(H, (6.24, 4.68)) → H'_2
  63. Translate(S, (6.24, 4.68)) → S'_6
  64. Translate(O'_6, (2.76, -3.68)) → F_1
  65. Translate(P'_7, (2.76, -3.68)) → K_1
  66. Translate(H'_2, (2.76, -3.68)) → L_1
  67. Translate(S'_6, (2.76, -3.68)) → M_1
  68. Translate(H, (2, 1.5)) → H'_4
  69. Translate(R, (2, 1.5)) → R'_1
  70. Translate(Q, (2, 1.5)) → Q'
  71. Translate(S, (2, 1.5)) → S'_7
  72. Translate(H, (2, 1.5)) → H'_5
  73. Translate(Q, (2, 1.5)) → Q'_1
  74. Translate(I, (2, 1.5)) → I'
  75. Translate(S, (6, 4.5)) → S'_8
  76. Translate(H, (6, 4.5)) → H'_6
  77. Translate(Q, (6, 4.5)) → Q'_2
  78. Translate(I, (6, 4.5)) → I'_1
  79. Translate(S'_8, (0, 0.5)) → N_1
  80. Translate(H'_6, (0, 0.5)) → O_1
  81. Translate(Q'_2, (0, 0.5)) → P_1
  82. Translate(I'_1, (0, 0.5)) → Q_1
  83. Translate(H'_4, (0, 0.5)) → R_1
  84. Translate(R'_1, (0, 0.5)) → S_1
  85. Translate(Q', (0, 0.5)) → Q''
  86. Translate(S, (6.24, (0.75 * 6.24))) → S'_{11}
  87. Translate(H, (6.24, (0.75 * 6.24))) → H'_{10}
  88. Translate(Q, (6.24, (0.75 * 6.24))) → Q'_5
  89. Translate(I, (6.24, (0.75 * 6.24))) → I'_3
  90. Translate(S, (6.24, (0.75 * 6.24))) → S'_{12}
  91. Translate(H, (6.24, (0.75 * 6.24))) → H'_{11}
  92. Translate(Q, (6.24, (0.75 * 6.24))) → Q'_6
  93. Translate(I, (6.24, (0.75 * 6.24))) → I'_4
  94. Translate(H, (2.24, 1.68)) → H'_{12}
  95. Translate(R, (2.24, 1.68)) → R'_4
  96. Translate(F, (2.24, 1.68)) → F'_2
  97. Translate(Q, (2.24, 1.68)) → Q'_7
  98. If(0 ≤ t ≤ 2.24, Polygon[Translate[P, (t, (0.75 * t))], Translate[G, (t, (0.75 * t))], Translate[R, (t, (0.75 * t))], Translate[H, (t, (0.75 * t))]]) → poly10
  99. Translate(P, (5, -2)) → P'_{10}
  100. Translate(G, (5, -2)) → G'_2
  101. Translate(R, (5, -2)) → R'_5
  102. Translate(H, (5, -2)) → H'_{13}
  103. Polygon(P'_{10}, G'_2, R'_5, H'_{13}) → poly6_f, p'_{10}, g'_2, r'_5, h'_{13}
  104. Translate(P, (2.24, 1.68)) → P'_{11}
  105. Translate(G, (2.24, 1.68)) → G'_3
  106. Translate(R, (2.24, 1.68)) → R'_6
  107. Translate(H, (2.24, 1.68)) → H'_{14}
  108. If(2.24 ≤ t ≤ 5, Polygon[Translate[P'_{11}, (t - 2.24, (-4 / 3 * (t - 2.24)))], Translate[G'_3, (t - 2.24, (-4 / 3 * (t - 2.24)))], Translate[R'_6, (t - 2.24, (-4 / 3 * (t - 2.24)))], Translate[H'_{14}, (t - 2.24, (-4 / 3 * (t - 2.24)))]]) → poly11
  109. Translate(O, (6.24, 4.68)) → O'_8
  110. Translate(P, (6.24, 4.68)) → P'
  111. Translate(H, (6.24, 4.68)) → H'
  112. Translate(S, (6.24, 4.68)) → S'_{13}
  113. If(5 ≤ t ≤ 11.24, Polygon[Translate[O, (t - 5, (0.75 * (t - 5)))], Translate[P, (t - 5, (0.75 * (t - 5)))], Translate[H, (t - 5, (0.75 * (t - 5)))], Translate[S, (t - 5, (0.75 * (t - 5)))]]) → poly12
  114. Translate(O, (6.24, 4.68)) → O'_9
  115. Translate(P, (6.24, 4.68)) → P'_8
  116. Translate(H, (6.24, 4.68)) → H'_{15}
  117. Translate(S, (6.24, 4.68)) → S'_{14}
  118. If(11.24 ≤ t ≤ 14, Polygon[Translate[O'_9, (t - 11.24, (-3.68 * (t - 11.24) / 2.76))], Translate[P'_8, (t - 11.24, (-3.68 * (t - 11.24) / 2.76))], Translate[H'_{15}, (t - 11.24, (-3.68 * (t - 11.24) / 2.76))], Translate[S'_{14}, (t - 11.24, (-3.68 * (t - 11.24) / 2.76))]]) → poly13
  119. Translate(O, (7, 3)) → O'_7
  120. Translate(P, (7, 3)) → P'_9
  121. Translate(H, (7, 3)) → H'_7
  122. Translate(S, (7, 3)) → S'_9
  123. Translate(O, (9, 1)) → O'_{10}
  124. Translate(P, (9, 1)) → P'_{12}
  125. Translate(S, (9, 1)) → S'_{15}
  126. Translate(O, (9, 1)) → O'_{11}
  127. Translate(P, (9, 1)) → P'_{13}
  128. Translate(S, (9, 1)) → S'_{16}
  129. Translate(O, (9, 1)) → O'_{12}
  130. Translate(P, (9, 1)) → P'_{14}
  131. Translate(H, (9, 1)) → H'_{18}
  132. Translate(S, (9, 1)) → S'_{17}
  133. Polygon(O'_{12}, P'_{14}, H'_{18}, S'_{17}) → poly5_f, o'_{12}, p'_{14}, h'_{18}, s'_{17}
  134. If(14 ≤ t ≤ 16.24, Polygon[Translate[H, (t - 14, (0.75 * (t - 14)))], Translate[R, (t - 14, (0.75 * (t - 14)))], Translate[F, (t - 14, (0.75 * (t - 14)))], Translate[Q, (t - 14, (0.75 * (t - 14)))]]) → poly15
  135. Translate(H, (2.24, 1.68)) → H'_{19}
  136. Translate(R, (2.24, 1.68)) → R'_2
  137. Translate(F, (2.24, 1.68)) → F'_3
  138. Translate(Q, (2.24, 1.68)) → Q'_8
  139. If(16.24 ≤ t ≤ 17.24, Polygon[Translate[H'_{19}, ((-0.24 * (t - 16.24)), (0.32 * (t - 16.24)))], Translate[R'_2, ((-0.24 * (t - 16.24)), (0.32 * (t - 16.24)))], Translate[F'_3, ((-0.24 * (t - 16.24)), (0.32 * (t - 16.24)))], Translate[Q'_8, ((-0.24 * (t - 16.24)), (0.32 * (t - 16.24)))]]) → poly14
  140. Translate(H, (2, 2)) → H'_{20}
  141. Translate(R, (2, 2)) → R'_7
  142. Translate(F, (2, 2)) → F'_4
  143. Translate(Q, (2, 2)) → Q'_9
  144. Polygon(H'_{20}, R'_7, F'_4, Q'_9) → poly8_f, h'_{20}, r'_7, f'_4, q'_9
  145. If(17.24 ≤ t ≤ 23.48, Polygon[Translate[S, (t - 17.24, (0.75 * (t - 17.24)))], Translate[H, (t - 17.24, (0.75 * (t - 17.24)))], Translate[Q, (t - 17.24, (0.75 * (t - 17.24)))], Translate[I, (t - 17.24, (0.75 * (t - 17.24)))]]) → poly16
  146. Translate(S, (6.24, 4.68)) → S'_{18}
  147. Translate(H, (6.24, 4.68)) → H'_8
  148. Translate(Q, (6.24, 4.68)) → Q'_3
  149. Translate(I, (6.24, 4.68)) → I'_5
  150. If(23.48 ≤ t ≤ 24.48, Polygon[Translate[S'_{18}, ((-0.24 * (t - 23.48)), (0.32 * (t - 23.48)))], Translate[H'_8, ((-0.24 * (t - 23.48)), (0.32 * (t - 23.48)))], Translate[Q'_3, ((-0.24 * (t - 23.48)), (0.32 * (t - 23.48)))], Translate[I'_5, ((-0.24 * (t - 23.48)), (0.32 * (t - 23.48)))]]) → poly17
  151. Translate(S, (6, 5)) → S'_{10}
  152. Translate(H, (6, 5)) → H'_9
  153. Translate(Q, (6, 5)) → Q'_4
  154. Translate(I, (6, 5)) → I'_2
  155. Polygon(S'_{10}, H'_9, Q'_4, I'_2) → poly7_f, s'_{10}, h'_9, q'_4, i'_2
  156. If(24.48 ≤ t ≤ 26.48, Translate[poly4, (t - 24.48, (2.5 * (t - 24.48)))]) → poly9
  157. Translate(poly4, (2, 5)) → poly4_f
  158. Polygon(F, G, 4) → poly18, c_1, d_1, e_1, j_1, A_1, B_1
  159. Polygon(J, F, 4) → poly19, k_1, l_1, m_1, n_1, G_1, T_1