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四边形面积探究：平行四边形、矩形、菱形与梯形的动态构造

四边形面积探究：平行四边形、矩形、菱形与梯形的动态构造
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2014-06-07 08:52
多边形角解析几何变换几何作图
所属专题 Deriving area formulas for Geometry Surface nets 6th Grade Deriving area formulas for Geometry
本范例通过GeoGebra动态演示了多种四边形（平行四边形、矩形、菱形、梯形）的面积关系与构造过程。核心数学概念包括四边形的定义、面积公式、对称性、旋转与镜像变换，以及集合论中图形分类的包含关系。作图要点涉及使用Line、OrthogonalLine、Point、Intersect、Segment、Midpoint、Semicircle、Rotate、Angle、Polygon、Mirror、Circle等命令，构造了多个点、线段、多边形及辅助圆，并通过旋转和镜像生成对称图形，最终形成两个多边形poly1与poly2，用于对比面积。适合中学几何教学，帮助学生直观理解不同四边形面积公式的推导与图形变换对面积的影响，也可用于探究面积守恒与等积变形问题。

GeoGebra 命令及参数

  1. Line(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Point(b) → C
  4. Line(C, a) → c
  5. OrthogonalLine(B, a) → d
  6. Intersect(c, d) → D
  7. Segment(A, B) → e
  8. Point(e) → E
  9. Segment(E, C) → f
  10. Segment(E, D) → g
  11. Midpoint(f) → F
  12. Midpoint(g) → G
  13. Semicircle(E, D) → h
  14. OrthogonalLine(E, e) → i
  15. Intersect(c, i) → H
  16. Semicircle(B, H) → k
  17. Semicircle(D, E) → p_1
  18. Semicircle(H, A) → q
  19. Semicircle(C, E) → r
  20. Semicircle(E, C) → s
  21. Rotate(E, -α, G) → E'
  22. Angle(E', G, E) → β
  23. Rotate(B, -α, G) → B'
  24. Angle(B', G, B) → γ
  25. Rotate(D, -α, G) → D'
  26. Angle(D', G, D) → δ
  27. Polygon(E', B', D') → poly1, d', e', b'
  28. Rotate(A, α, F) → A'
  29. Angle(A, F, A') → ε
  30. Rotate(E, α, F) → E'_1
  31. Angle(E, F, E'_1) → ζ
  32. Rotate(C, α, F) → C'
  33. Angle(C, F, C') → η
  34. Polygon(A', C', E'_1) → poly2, e'_1, a', c'
  35. Midpoint(e) → I
  36. Midpoint(A, C) → J
  37. Midpoint(C, D) → K
  38. Midpoint(B, D) → L
  39. Polygon(J, I, L, K) → poly3, j_1, i_1, l, k_1
  40. Midpoint(j_1) → M
  41. Midpoint(i_1) → N
  42. Midpoint(l) → O
  43. Midpoint(k_1) → P
  44. Rotate(J, α, P) → J'
  45. Angle(J, P, J') → θ
  46. Rotate(C, α, P) → C'_1
  47. Angle(C, P, C'_1) → ι
  48. Rotate(K, α, P) → K'
  49. Angle(K, P, K') → κ
  50. Polygon(J', C'_1, K') → poly4, k', j', c'_1
  51. Rotate(K, -α, O) → K'_1
  52. Angle(K'_1, O, K) → λ
  53. Rotate(L, -α, O) → L'
  54. Angle(L', O, L) → μ
  55. Rotate(D, -α, O) → D'_1
  56. Angle(D'_1, O, D) → ν
  57. Polygon(K'_1, D'_1, L') → poly5, l', k'_1, d'_1
  58. Rotate(A, -α, M) → A'_1
  59. Angle(A'_1, M, A) → ξ
  60. Rotate(I, -α, M) → I'
  61. Angle(I', M, I) → ο
  62. Rotate(J, -α, M) → J'_1
  63. Angle(J'_1, M, J) → ρ
  64. Polygon(J'_1, A'_1, I') → poly6, i', j'_1, a'_1
  65. Rotate(B, α, N) → B'_1
  66. Angle(B, N, B'_1) → σ
  67. Rotate(L, α, N) → L'_1
  68. Angle(L, N, L'_1) → τ
  69. Rotate(I, α, N) → I'_1
  70. Angle(I, N, I'_1) → υ
  71. Polygon(B'_1, L'_1, I'_1) → poly7, i'_1, b'_1, l'_1
  72. Segment(A', C') → n_2
  73. Polygon(C, E, D) → poly8, d_1, c_1, e_1
  74. Midpoint(A, C) → Q
  75. Midpoint(D, C) → R
  76. Midpoint(A, B) → S
  77. Midpoint(B, D) → T
  78. Segment(Q, T) → t
  79. Segment(S, R) → a_1
  80. Intersect(t, a_1) → U
  81. Point(e) → V
  82. Mirror(V, U) → V'
  83. Polygon(C, Q, V, B, T, V') → poly9, c_2, q_1, v, b_1, t_1, v'
  84. Rotate(A, α, Q) → A'_2
  85. Angle(A, Q, A'_2) → ϕ
  86. Rotate(V, α, Q) → V'_1
  87. Angle(V, Q, V'_1) → χ
  88. Polygon(Q, V'_1, A'_2) → poly10, a'_2, q_2, v'_1
  89. Rotate(V', α, T) → V''
  90. Angle(V', T, V'') → ψ
  91. Rotate(D, α, T) → D'_2
  92. Angle(D, T, D'_2) → ω
  93. Polygon(V'', D'_2, T) → poly11, t_2, v'', d'_2
  94. Segment(C, D) → f_1
  95. Segment(C, V') → g_1
  96. Segment(V, B) → h_1
  97. Segment(Q, C) → l_1
  98. Midpoint(A, C) → W
  99. Midpoint(B, D) → Z
  100. Segment(V, B) → m_1
  101. Point(m_1) → A_1
  102. Polygon(W, C, D, Z, A_1, V) → poly12, w, c_3, d_2, z, u_2, v_1
  103. Rotate(A_1, -α, Z) → B_1
  104. Angle(B_1, Z, A_1) → α_1
  105. Rotate(B, -α, Z) → B'_2
  106. Angle(B'_2, Z, B) → β_1
  107. Polygon(B_1, Z, B'_2) → poly13, b'_2, b_2, z_1
  108. Segment(E, C) → n_1
  109. Segment(C, D) → r_1
  110. Segment(E, D) → s_1
  111. Midpoint(s_1) → C_1
  112. Rotate(E, α, C_1) → E'_2
  113. Angle(E, C_1, E'_2) → γ_1
  114. Rotate(C, α, C_1) → C'_2
  115. Angle(C, C_1, C'_2) → δ_1
  116. Rotate(D, α, C_1) → D'_3
  117. Angle(D, C_1, D'_3) → ε_1
  118. Polygon(E'_2, C'_2, D'_3) → poly14, d'_3, e'_2, c'_2
  119. Midpoint(A, C) → D_1
  120. Midpoint(B, D) → E_1
  121. Mirror(V, D_1) → V'_2
  122. Mirror(A_1, E_1) → F_1
  123. Polygon(V, V'_2, F_1, A_1) → poly15, v_2, v'_2, f_2, a_2
  124. Midpoint(A_1, F_1) → G_1
  125. Rotate(A_1, α, G_1) → H_1
  126. Angle(A_1, G_1, H_1) → ζ_1
  127. Rotate(V, α, G_1) → V'_3
  128. Angle(V, G_1, V'_3) → η_1
  129. Rotate(V'_2, α, G_1) → I_1
  130. Angle(V'_2, G_1, I_1) → θ_1
  131. Rotate(F_1, α, G_1) → J_1
  132. Angle(F_1, G_1, J_1) → ι_1
  133. Polygon(V'_3, I_1, J_1, H_1) → poly16, v'_3, i_2, j_2, h_2
  134. Polygon(E'_1, A', C') → poly17, c'_3, e'_3, a'_3
  135. Polygon(A, C, D, B) → poly18, a_3, c_4, d_3, b_3
  136. Polygon(C, E, D) → poly19, d_4, c_5, e_2
  137. Polygon(E', B', D') → poly20, d'_4, e'_4, b'_3
  138. Polygon(A'_1, J'_1, I') → poly21, i'_2, a'_4, j'_2
  139. Polygon(B'_1, I'_1, L'_1) → poly22, l'_2, b'_4, i'_3
  140. Polygon(J', K', C'_1) → poly23, c'_4, j'_3, k'_2
  141. Polygon(K'_1, D'_1, L') → poly24, l'_3, k'_3, d'_5
  142. Polygon(A, C, D, B) → poly25, a_4, c_6, d_5, b_4
  143. Polygon(I, J, K, L) → poly26, i_3, j_3, k_2, l_2
  144. Polygon(V'_1, A'_2, Q) → poly27, q_3, v'_4, a'_5
  145. Polygon(D'_2, T, V'') → poly28, v''_1, d'_6, t_3
  146. Polygon(V, V'_1, V', V'') → poly29, v_3, v'_5, v'_6, v''_2
  147. Polygon(A, C, D, B) → poly30, a_5, c_7, d_6, b_5
  148. Point(b_5) → K_1
  149. Polygon(V'_1, A'_2, Q) → poly31, q_4, v'_7, a'_6
  150. Polygon(B_1, Z, B'_2) → poly32, b'_5, b_6, z_2
  151. Polygon(V, V'_1, B_1, A_1) → poly33, v_4, v'_8, b_7, a_6
  152. Polygon(A, C, D, B) → poly34, a_7, c_8, d_7, b_8
  153. Polygon(E, C, D) → poly35, d_8, e_3, c_9
  154. Polygon(D'_3, E'_2, C'_2) → poly36, c'_5, d'_7, e'_5
  155. Polygon(C'_2, E, C, D) → poly37, c'_6, e_4, c_{10}, d_9
  156. Polygon(V, V'_2, F_1, A_1) → poly38, v_5, v'_9, f_3, a_8
  157. Polygon(J_1, H_1, V'_3, I_1) → poly39, j_4, h_3, v'_{10}, i_4
  158. Polygon(V, V'_2, V'_3, I_1) → poly40, v_6, v'_{11}, v'_{12}, i_5
  159. Midpoint(A, E) → L_1
  160. Midpoint(E, B) → M_1
  161. Mirror(C, L_1) → C'_3
  162. Polygon(A, C, D, B) → poly41, a_9, c_{11}, d_{10}, b_9
  163. Polygon(L_1, C, D, M_1) → poly42, l_3, c_{12}, d_{11}, m_2
  164. Mirror(A, L_1) → A'_3
  165. Mirror(A'_3, L_1) → O_1
  166. Intersect(l_3, c_{12}) → P_1
  167. Rotate(O_1, -α, L_1) → Q_1
  168. Angle(Q_1, L_1, O_1) → κ_1
  169. Rotate(P_1, -α, L_1) → R_1
  170. Angle(R_1, L_1, P_1) → λ_1
  171. Polygon(Q_1, R_1, L_1) → poly43, l_4, q_5, r_2
  172. Rotate(D, α, M_1) → D'_4
  173. Angle(D, M_1, D'_4) → μ_1
  174. Rotate(B, α, M_1) → B'_3
  175. Angle(B, M_1, B'_3) → ν_1
  176. Polygon(M_1, B'_3, D'_4) → poly44, d'_8, m_3, b'_6
  177. Polygon(R_1, C, D) → poly45, d_{12}, r_3, c_{13}
  178. Circle(C, D) → p_2
  179. Line(A, B) → g_2
  180. Intersect(p_2, g_2) → N_1, S_1
  181. Polygon(A, C, D, B) → poly46, a_{10}, c_{14}, d_{13}, b_{10}
  182. Polygon(C, S_1, B, D) → poly47, c_{15}, s_2, b_{11}, d_{14}
  183. Segment(A, B) → g_3
  184. Midpoint(g_3) → T_1
  185. Rotate(A, -α, T_1) → A'_4
  186. Angle(A'_4, T_1, A) → ξ_1
  187. Segment(C, D) → k_3
  188. Midpoint(c_{14}) → U_1
  189. Rotate(C, -α, U_1) → C'_4
  190. Angle(C'_4, U_1, C) → ο_1
  191. Midpoint(S_1, B) → V_1
  192. Mirror(A, V_1) → A'_5
  193. Midpoint(S_1, A'_5) → W_1
  194. Rotate(S_1, -α, W_1) → Z_1
  195. Angle(Z_1, W_1, S_1) → ρ_1
  196. Polygon(A'_4, C'_4, Z_1) → poly48, z_3, a'_7, c'_7
  197. Polygon(C, S_1, Z_1, D) → poly49, c_{16}, s_3, z_4, d_{15}

AI 绘图提示语参考

画一个动态的几何图形，展示平行四边形、矩形、菱形和梯形的面积关系与构造。首先绘制一条线段和它的垂线，得到一个直角三角形。以线段为直径画半圆，交点连接形成平行四边形。再以线段中点为中心，旋转半圆得到另一个平行四边形，用于面积比较。最后，连接各点形成多个多边形，并旋转部分多边形展示对称性和面积变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。