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毕达哥拉斯定理——面积分割

毕达哥拉斯定理——面积分割
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2017-12-11 15:54
测试改名0解析几何几何作图
该范例演示了毕达哥拉斯定理的一种面积分割证明方法。核心数学概念包括面积、毕达哥拉斯定理及其证明。通过构造中点、圆、线段、旋转、交点等，展示了面积分割的几何作图过程。适合初中及以上学生学习几何证明。

GeoGebra 命令及参数

  1. Midpoint(A, C) → A_0
  2. Circle(A_0, C) → c
  3. Point(c) → B
  4. Segment(A, C) → a
  5. Segment(C, B) → b
  6. Segment(B, A) → d
  7. Rotate(B, 90°, C) → B_2
  8. Rotate(C, -90°, B) → B_1
  9. Rotate(B, -90°, A) → A_2
  10. Rotate(A, 90°, B) → A_1
  11. Rotate(A, -90°, C) → C_2
  12. Rotate(C, 90°, A) → C_1
  13. Segment(A, A_2) → e
  14. Segment(A_2, A_1) → f
  15. Segment(A_1, B) → g
  16. Segment(B, B_1) → h
  17. Segment(B_1, B_2) → i
  18. Segment(B_2, C) → j
  19. Segment(A, C_1) → k
  20. Segment(C_1, C_2) → l
  21. Segment(C_2, C) → m
  22. Midpoint(A, B_1) → D
  23. Line(D, a) → n
  24. Intersect(n, j) → E
  25. Midpoint(D, E) → F
  26. OrthogonalLine(F, n) → p_4
  27. Intersect(p_4, b) → G
  28. Intersect(p_4, i) → H
  29. Segment(D, E) → q_3
  30. Segment(H, G) → r_3
  31. Midpoint(C, A_1) → I
  32. Line(I, a) → s_3
  33. Intersect(s_3, e) → J
  34. Midpoint(J, I) → K
  35. OrthogonalLine(K, s_3) → t_3
  36. Segment(J, I) → a_1
  37. Intersect(t_3, f) → L
  38. Intersect(t_3, d) → M
  39. Segment(L, M) → b_1
  40. Distance(A, B) → AB
  41. Distance(B, C) → BC
  42. Min(AB, BC) → MinS
  43. If(BC > AB, Distance[D, F], Distance[L, K]) → Len2
  44. Circle(C, Len2) → c_1
  45. Circle(C_1, Len2) → d_1
  46. Intersect(c_1, m, 1) → N_1
  47. Intersect(c_1, a, 1) → N_2
  48. Intersect(d_1, l, 1) → N_4
  49. Intersect(d_1, k, 1) → N_3
  50. Line(N_1, b) → e_1
  51. Line(N_2, d) → f_1
  52. Line(N_4, d) → g_1
  53. Line(N_3, b) → h_1
  54. Intersect(e_1, f_1) → O_1
  55. Intersect(e_1, g_1) → O_2
  56. Intersect(g_1, h_1) → O_3
  57. Intersect(f_1, h_1) → O_4
  58. Segment(N_1, O_1) → i_1
  59. Segment(N_2, O_4) → j_1
  60. Segment(N_3, O_3) → k_1
  61. Segment(N_4, O_2) → l_1
  62. Circle(C, Len2) → p_1
  63. Circle(C_1, Len2) → q_1
  64. Intersect(q_1, k, 1) → M_4
  65. Intersect(q_1, l, 1) → M_3
  66. Intersect(p_1, m, 1) → M_2
  67. Intersect(p_1, a, 1) → M_1
  68. Line(M_1, b) → m_1
  69. Line(M_3, b) → n_1
  70. Line(M_4, h) → r_1
  71. Line(M_2, j) → s_1
  72. Intersect(n_1, r_1) → P_4
  73. Intersect(n_1, s_1) → P_3
  74. Intersect(m_1, s_1) → P_2
  75. Intersect(m_1, r_1) → P_1
  76. Segment(M_3, P_4) → t_1
  77. Segment(M_4, P_1) → a_2
  78. Segment(M_1, P_2) → b_2
  79. Segment(M_2, P_3) → c_2
  80. Circle(Q_1, Len2) → d_2
  81. Line(Q_1, m) → e_2
  82. Line(Q_1, a) → f_2
  83. Intersect(d_2, f_2, 1) → Q_2
  84. Intersect(d_2, e_2, 1) → Q_3
  85. Line(Q_3, b) → g_2
  86. Line(Q_2, b) → h_2
  87. Line(Q_2, d) → i_2
  88. Line(Q_3, d) → j_2
  89. Intersect(g_2, i_2) → Q_4
  90. Intersect(h_2, j_2) → Q_5
  91. If(BC > AB, Polygon[Q_1, Q_3, Q_4, Q_2], Polygon[Q_1, Q_2, Q_5, Q_3]) → RedPoly
  92. Circle(R_1, Len2) → k_2
  93. Line(R_1, a) → l_2
  94. Line(R_1, m) → m_2
  95. Intersect(k_2, l_2, 1) → R_2
  96. Intersect(k_2, m_2, 2) → R_3
  97. Line(R_2, d) → n_2
  98. Line(R_3, d) → p_2
  99. Line(R_2, b) → q_2
  100. Line(R_3, b) → r_2
  101. Intersect(p_2, q_2) → R_4
  102. Intersect(n_2, r_2) → R_5
  103. If(BC > AB, (Polygon[R_1, R_2, R_5, R_3] * Polygon[R_1, R_3, R_4, R_5])) → o
  104. If(BC > AB, Polygon[R_1, R_2, R_5, R_3], Polygon[R_1, R_3, R_4, R_2]) → BluePoly
  105. Circle(S_1, Len2) → s_2
  106. Line(S_1, a) → t_2
  107. Line(S_1, m) → a_3
  108. Intersect(s_2, a_3, 2) → S_2
  109. Intersect(s_2, t_2, 2) → S_3
  110. Line(S_2, b) → b_3
  111. Line(S_3, b) → c_3
  112. Line(S_2, d) → d_3
  113. Line(S_3, d) → e_3
  114. Intersect(c_3, d_3) → S_5
  115. Intersect(b_3, e_3) → S_4
  116. If(BC > AB, Polygon[S_1, S_2, S_4, S_3], Polygon[S_1, S_3, S_5, S_2]) → AmberPoly
  117. Circle(T_1, Len2) → f_3
  118. Line(T_1, a) → g_3
  119. Line(T_1, m) → h_3
  120. Intersect(f_3, h_3, 1) → T_2
  121. Intersect(f_3, g_3, 2) → T_3
  122. Line(T_2, d) → i_3
  123. Line(T_3, d) → j_3
  124. Line(T_3, b) → k_3
  125. Line(T_2, b) → l_3
  126. Intersect(j_3, l_3) → T_4
  127. Intersect(i_3, k_3) → T_5
  128. If(BC > AB, Polygon[T_1, T_3, T_4, T_2], Polygon[T_1, T_2, T_5, T_3]) → GreenPoly
  129. Circle(U_1, MinS) → p_3
  130. Line(U_1, g) → m_3
  131. Intersect(p_3, m_3, 2) → U_2
  132. Rotate(U_2, 90°, U_1) → U_3
  133. Rotate(U_1, -90°, U_2) → U_4
  134. Polygon(U_3, U_1, U_2, U_4) → poly1, u_3, u_1, u_2, u_4
  135. Vector(Q_1, C) → p
  136. Vector(R_1, C_2) → q
  137. Vector(S_1, C_1) → r
  138. Vector(T_1, A) → s
  139. Translate(RedPoly, p_t) → RedPoly'
  140. Translate(BluePoly, q_t) → BluePoly'
  141. Translate(AmberPoly, r_t) → AmberPoly'
  142. Translate(GreenPoly, s_t) → GreenPoly'
  143. Vector(U_1, O_1) → u
  144. Vector(U_1, P_1) → v
  145. If(BC > AB, Vector[(Brown * u)], Vector[(Brown * v)]) → u_t
  146. Translate(U_3, u_t) → N
  147. Translate(U_1, u_t) → O
  148. Translate(U_2, u_t) → P
  149. Translate(U_4, u_t) → Q
  150. Polygon(N, O, P, Q) → poly1', n_3, o_1, p_5, q_4
  151. Polygon(A, B, C) → poly2, c_4, a_4, b_4
  152. Circle(B, S90) → d_4
  153. Intersect(d_4, b, 1) → R
  154. Intersect(d_4, d, 1) → S
  155. Rotate(B, -90°, R) → B'
  156. Segment(S, B') → e_4
  157. Segment(B', R) → f_4
  158. Translate(RedPoly, v_1) → RedPoly'_1
  159. Translate(BluePoly, v_2) → BluePoly'_1
  160. Translate(AmberPoly, v_3) → AmberPoly'_1
  161. Translate(GreenPoly, v_4) → GreenPoly'_1
  162. If(BC > AB, Vector[(Rearrange * u)], Vector[(Rearrange * v)]) → v_t
  163. Translate(U_3, v_t) → T
  164. Translate(U_1, v_t) → U
  165. Translate(U_2, v_t) → V
  166. Translate(U_4, v_t) → W
  167. Polygon(T, U, V, W) → poly1'_1, t_4, u_5, v_5, w_1
  168. SelectedIndex(Bahasa) → nn

AI 绘图提示语参考

绘制一个直角三角形ABC，其中A、B、C分别为直角顶点、斜边顶点和底边顶点。通过构造中点、圆、旋转、线段和交点，展示毕达哥拉斯定理的面积分割证明。首先画出A、C两点，然后画出AC线段，接着构造AC的中点A0，以A0为圆心画圆c，圆c与AC交于B点。再构造AB、BC、CA线段，通过旋转和交点构造一系列辅助线段和点，最终通过这些辅助线段和点证明毕达哥拉斯定理。
以上提示语为根据原始绘图命令整理而来，仅供参考。