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勾股定理的面积剖分演示（一）

勾股定理的面积剖分演示（一）
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2017-12-11 15:53
测试改名0多边形变换几何作图
所属专题 Mathematics Form 1
① 本范例通过将直角三角形三边上的正方形进行面积剖分与重组，直观展示勾股定理中两直角边平方和等于斜边平方的几何关系。② 核心数学概念是勾股定理（Pythagorean Theorem）以及面积守恒的剖分证明方法，涉及平面图形（三角形、正方形、半圆）的面积计算与比较。③ 作图要点：从构造摘要可见，使用了半圆（Semicircle）、中点（Midpoint）、点（Point）、线段（Segment）、旋转（Rotate）、射线（Ray）、相交（Intersect）、直线（Line）、垂线（OrthogonalLine）、距离（Distance）、最大值/最小值（Max/Min）、圆（Circle）、多边形（Polygon）、向量（Vector）、平移（Translate）、角（Angle）等GeoGebra命令，通过构造辅助线、剖分区域并动态调整，实现面积的重组与对比。④ 适合初中几何教学或数学科普，用于帮助学生理解勾股定理的几何证明，也可作为动态几何课件设计的参考。

GeoGebra 命令及参数

  1. Semicircle(C, A) → c
  2. Midpoint(A, C) → C1
  3. Point(c) → B
  4. Segment(A, B) → a
  5. Segment(B, C) → b
  6. Segment(A, C) → d
  7. Rotate(A, 90°, B) → A2
  8. Rotate(B, -90°, A) → A1
  9. Segment(A, A1) → e
  10. Segment(A1, A2) → f
  11. Segment(A2, B) → g
  12. Rotate(B, 90°, C) → B2
  13. Rotate(C, -90°, B) → B1
  14. Segment(B, B1) → h
  15. Segment(B1, B2) → i
  16. Segment(B2, C) → j
  17. Rotate(A, -90°, C) → C2
  18. Rotate(C, 90°, A) → C3
  19. Segment(A, C3) → k
  20. Segment(C3, C2) → l
  21. Segment(C2, C) → m
  22. Rotate(A, -45°, B) → A'
  23. Ray(B, A') → n
  24. Intersect(d, n) → D
  25. Line(D, a) → p
  26. Line(D, b) → q
  27. Intersect(q, a) → A3
  28. Intersect(p, b) → B3
  29. Line(A3, k) → r
  30. Line(B3, m) → s
  31. OrthogonalLine(B3, s) → t
  32. Intersect(s, h) → B6
  33. Intersect(t, j) → B4
  34. Intersect(r, g) → A4
  35. OrthogonalLine(A4, r) → c_1
  36. OrthogonalLine(A3, r) → d_1
  37. Intersect(d_1, e) → A6
  38. Intersect(c_1, f) → A5
  39. OrthogonalLine(B6, s) → a_1
  40. Intersect(a_1, i) → B5
  41. Segment(A4, A5) → f_1
  42. Segment(A5, A6) → g_1
  43. Segment(B3, B4) → h_1
  44. Segment(B4, B5) → i_1
  45. Distance(A4, A5) → LenS1
  46. Distance(B4, B5) → LenS2
  47. Max(LenS1, LenS2) → BLen
  48. Min(LenS1, LenS2) → SLen
  49. Circle(A, LenS1) → q_1
  50. Intersect(q_1, k, 1) → C5
  51. Intersect(q_1, d, 1) → C8
  52. Line(C5, d) → l_1
  53. Line(C8, k) → m_1
  54. Intersect(m_1, l) → C7
  55. Intersect(l_1, m) → C6
  56. Intersect(l_1, m_1) → C4
  57. Line(C3, C4) → p_1
  58. Line(C, p_1) → r_1
  59. OrthogonalLine(C7, p_1) → n_1
  60. Line(C5, n_1) → s_1
  61. Segment(C8, C6) → t_1
  62. Intersect(p_1, t_1) → C10
  63. Intersect(r_1, t_1) → C9
  64. Intersect(p_1, n_1) → C11
  65. Intersect(p_1, s_1) → C12
  66. Segment(C3, C4) → a_2
  67. Segment(C4, C10) → b_2
  68. Segment(C, C9) → c_2
  69. Segment(C11, C7) → h_2
  70. Segment(C5, C12) → i_2
  71. Distance(A, A3) → LenT1
  72. Distance(A, A6) → LenT2
  73. Distance(B, B6) → LenT4
  74. Distance(B, B3) → LenT3
  75. Line(H, a) → s_2
  76. Line(H, b) → t_2
  77. Circle(H, LenT1) → c_3
  78. Circle(H, LenT2) → d_3
  79. Intersect(c_3, t_2, 2) → H2
  80. Intersect(d_3, s_2, 2) → H1
  81. Polygon(H, H1, H2) → poly4, h2, h_3, h1
  82. Line(I, a) → a_3
  83. Line(I, b) → b_3
  84. Circle(I, LenT1) → k_3
  85. Circle(I, LenT2) → p_3
  86. Intersect(p_3, b_3, 2) → I1
  87. Intersect(k_3, a_3, 1) → I2
  88. Polygon(I, I1, I2) → poly5, i2, i_3, i1
  89. Line(K, a) → m_3
  90. Line(K, b) → n_3
  91. Circle(K, LenT3) → s_3
  92. Circle(K, LenT4) → t_3
  93. Intersect(t_3, m_3, 1) → K2
  94. Intersect(s_3, n_3, 1) → K1
  95. Polygon(K, K2, K1) → poly7, k1, k_4, k2
  96. Line(L, a) → a_4
  97. Line(L, b) → b_4
  98. Circle(L, LenT3) → c_4
  99. Circle(L, LenT4) → d_4
  100. Intersect(c_4, b_4, 2) → L3
  101. Intersect(d_4, a_4, 2) → L2
  102. Line(M, a) → e_4
  103. Line(M, b) → f_4
  104. Circle(M, LenT3) → h_4
  105. Circle(M, LenT4) → p_4
  106. Intersect(h_4, e_4, 1) → M2
  107. Intersect(p_4, f_4, 2) → M1
  108. Line(N, a) → i_4
  109. Line(N, b) → n_4
  110. Circle(N, LenT3) → q_4
  111. Circle(N, LenT4) → r_4
  112. Intersect(q_4, i_4, 2) → N2
  113. Intersect(r_4, n_4, 1) → N1
  114. Polygon(N, N1, N2) → poly10, n2, n_5, n1
  115. Vector(I, C11) → u3
  116. Translate(I, v3) → I'
  117. Translate(I1, v3) → I1'
  118. Translate(I2, v3) → I2'
  119. Polygon(I', I1', I2') → poly5', i2', i', i1'
  120. Vector(H, C9) → u4
  121. Vector(K, C12) → u6
  122. Translate(K, v6) → K'
  123. Translate(K2, v6) → K2'
  124. Translate(K1, v6) → K1'
  125. Polygon(K', K2', K1') → poly7', k1', k', k2'
  126. Vector(N, C9) → u9
  127. Translate(N, v9) → N'
  128. Translate(N1, v9) → N1'
  129. Translate(N2, v9) → N2'
  130. Polygon(N', N1', N2') → poly10', n2', n', n1'
  131. Translate(H, v5) → H'
  132. Translate(H1, v5) → H1'
  133. Translate(H2, v5) → H2'
  134. Polygon(H', H1', H2') → poly4', h2', h', h1'
  135. OrthogonalLine(E, b) → b_1
  136. OrthogonalLine(E, a) → e_1
  137. Circle(E, LenT1) → k_1
  138. Circle(E, Distance[A, B]) → d_2
  139. Intersect(k_1, e_1, 1) → E1
  140. Intersect(d_2, b_1, 2) → E2
  141. Line(E2, b) → j_1
  142. Circle(E2, LenT2) → e_2
  143. Intersect(e_2, j_1, 1) → E3
  144. Line(E3, m) → f_2
  145. Line(E1, d) → g_2
  146. Intersect(f_2, g_2) → E4
  147. Polygon(E, E1, E4, E3, E2) → poly1, e_3, e1, g_3, f_3, e2
  148. Line(F, b) → j_2
  149. Line(F, a) → k_2
  150. Circle(F, Distance[B, C]) → p_2
  151. Intersect(p_2, k_2, 1) → F1
  152. Circle(F, LenT4) → q_2
  153. Intersect(q_2, j_2, 2) → F3
  154. Line(F1, b) → l_2
  155. Circle(F1, LenT3) → r_2
  156. Intersect(r_2, l_2, 2) → F4
  157. Line(F4, d) → m_2
  158. Line(F3, k) → n_2
  159. Intersect(m_2, n_2) → F5
  160. Polygon(F, F1, F4, F5, F3) → poly2, f_5, f1, f4, f5, f3
  161. Vector(E, C10) → u1
  162. Translate(E, v1) → E'
  163. Translate(E1, v1) → E1'
  164. Translate(E4, v1) → E4'
  165. Translate(E3, v1) → E3'
  166. Translate(E2, v1) → E2'
  167. Polygon(E', E1', E4', E3', E2') → poly1', e', e1', e4', e3', e2'
  168. Vector(F, C10) → u2
  169. Translate(F, v2) → F'
  170. Translate(F1, v2) → F1'
  171. Translate(F4, v2) → F4'
  172. Translate(F5, v2) → F5'
  173. Translate(F3, v2) → F3'
  174. Polygon(F', F1', F4', F5', F3') → poly2', f', f1', f4', f5', f3'
  175. Circle(B, S90) → q_3
  176. Intersect(q_3, a, 1) → G
  177. Intersect(q_3, b, 1) → J
  178. Rotate(B, -90°, J) → B'
  179. Translate(E, w1) → E'_1
  180. Translate(E1, w1) → E1'_1
  181. Translate(E4, w1) → E4'_1
  182. Translate(E3, w1) → E3'_1
  183. Translate(E2, w1) → E2'_1
  184. Polygon(E'_1, E1'_1, E4'_1, E3'_1, E2'_1) → poly1'_1, e'_1, e1'_1, e4'_1, e3'_1, e2'_1
  185. Translate(F, w2) → F'_1
  186. Translate(F1, w2) → F1'_1
  187. Translate(F4, w2) → F4'_1
  188. Translate(F5, w2) → F5'_1
  189. Translate(F3, w2) → F3'_1
  190. Polygon(F'_1, F1'_1, F4'_1, F5'_1, F3'_1) → poly2'_1, f'_1, f1'_1, f4'_1, f5'_1, f3'_1
  191. Translate(I, w3) → I'_1
  192. Translate(I1, w3) → I1'_1
  193. Translate(I2, w3) → I2'_1
  194. Polygon(I'_1, I1'_1, I2'_1) → poly5'_1, i2'_1, i'_1, i1'_1
  195. Translate(H, w4) → H'_1
  196. Translate(H1, w4) → H1'_1
  197. Translate(H2, w4) → H2'_1
  198. Polygon(H'_1, H1'_1, H2'_1) → poly4'_1, h2'_1, h'_1, h1'_1
  199. Translate(K, w6) → K'_1
  200. Translate(K2, w6) → K2'_1
  201. Translate(K1, w6) → K1'_1
  202. Polygon(K'_1, K2'_1, K1'_1) → poly7'_1, k1'_1, k'_1, k2'_1
  203. Translate(N, w9) → N'_1
  204. Translate(N1, w9) → N1'_1
  205. Translate(N2, w9) → N2'_1
  206. Polygon(N'_1, N1'_1, N2'_1) → poly10'_1, n2'_1, n'_1, n1'_1
  207. Angle(C, B, A) → α
  208. SelectedIndex(Bahasa) → nn