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GoGeometry 行动24：双正方形定理的动态演示

GoGeometry 行动24：双正方形定理的动态演示
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2017-04-14 08:22
多边形角变换几何作图滑块动画
① 本范例动态演示了“双正方形定理”的更多现象，通过可拖拽的几何构造直观展示两个正方形组合后产生的线段、角度与面积关系。② 核心数学概念包括几何变换（旋转、反射、平移）、平行四边形性质、中点与垂线关系，以及圆与圆弧的辅助构造，涉及LCM与GCD的趣味联系。③ 作图要点：使用GeoGebra命令如Ray、Segment、Circle、Intersect、Point、OrthogonalLine、Line、Polygon、Rotate、Dilate、Midpoint、CircleArc、CircleSector、Vector、Translate、Angle、Mirror等，构建了多个点、线、多边形及圆弧对象，如A、B、C、D、poly1、圆与辅助线，通过滑块或拖拽实现动态变化。④ 适合中学几何教学、数学竞赛辅导或几何探究活动，帮助学生理解变换与定理的直观证明，也可用于教师制作交互式课件。

GeoGebra 命令及参数

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Point(n_2) → K_1
  7. Segment(B_2, K_1) → r_2
  8. OrthogonalLine(K_1, r_2) → s_2
  9. Circle(K_1, r_2) → t_2
  10. Intersect(t_2, s_2, 2) → L_1
  11. Line(K_1, L_1) → h_3
  12. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  13. Segment(B_2, C) → h
  14. OrthogonalLine(L_1, h) → f_3
  15. Intersect(f_3, h) → M_1
  16. Line(K_1, M_1) → i_2
  17. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  18. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  19. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  20. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  21. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  22. Segment(R, A''') → f''
  23. Intersect(i_2, f'') → V_1
  24. Dilate(M_1, If[16 < e ≤ 17, 17 - e, 0], V_1) → Z_1
  25. Segment(M_1, Z_1) → e_2
  26. Line(C, e_2) → f_4
  27. Dilate(A, If[17 < e ≤ 18, 18 - e, 0], M_1) → A'_4
  28. Midpoint(A, C) → V
  29. CircleArc(V, C, D) → g_2
  30. Point(g_2) → B
  31. Segment(C, B) → j_1
  32. Segment(A, B) → n
  33. Intersect(n, f'') → Z
  34. Segment(B, Z) → l_1
  35. Segment(A, A') → f
  36. Segment(A'_1, A'') → f'
  37. Segment(C, D) → i
  38. Circle(B_2, (0.2 * r_2)) → p
  39. Segment(D, A) → j
  40. Circle(C, Radius[p]) → k
  41. Circle(A, (0.1 * j)) → q
  42. Circle(D, (0.1 * j)) → r
  43. Segment(A, C) → l
  44. Segment(D, B_2) → m
  45. Intersect(r, j, 1) → F
  46. Intersect(r, m, 1) → G
  47. Intersect(r, i, 1) → H
  48. Intersect(k, i, 1) → I
  49. Intersect(k, l, 1) → J
  50. Intersect(k, h, 1) → K
  51. Intersect(p, h, 1) → L
  52. Intersect(p, m, 1) → M
  53. Intersect(p, g, 1) → N
  54. Intersect(q, g, 1) → O
  55. Intersect(q, l, 1) → P
  56. Intersect(q, j, 1) → Q
  57. CircleSector(B_2, L, N) → s
  58. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  59. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  60. Segment(R', S) → f'''
  61. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  62. Vector(B_2, B') → u
  63. Translate(s, u) → s'
  64. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  65. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  66. Vector(C, C') → v
  67. Translate(s'', v) → s'''
  68. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  69. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  70. Vector(D, D') → w
  71. Translate(t, w) → t'
  72. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  73. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  74. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  75. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  76. Vector(B_2, B'_1) → v_1
  77. Vector(B_2, B'_3) → w_1
  78. Vector(B_2, B'_2) → i_1
  79. Circle(A, n - j_1) → h_2
  80. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  81. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  82. Vector(B_2, B'_4) → u_1
  83. Translate(s, u_1) → s'_1
  84. Line(B, f') → a_1
  85. Point(a_1) → U
  86. Angle(U, B, C) → α
  87. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  88. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  89. Line(A, B_2) → h_1
  90. Angle(B, A, D) → β
  91. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  92. Mirror(C, D) → C'_1
  93. Rotate(C, ((-π)) / 3, B_2) → C'_2
  94. Segment(B_2, C'_2) → f_1
  95. Segment(C, C'_2) → g_1
  96. Intersect(k, g_1, 1) → E
  97. Intersect(k, h) → T,
  98. CircleSector(C, T, E) → d_1
  99. Rotate(D, 15°, C) → D'_2
  100. Angle(D, C, D'_2) → γ
  101. Ray(C, D'_2) → k_1
  102. Rotate(C, (-(15°)), D) → C'_3
  103. Angle(C'_3, D, C) → δ
  104. Ray(D, C'_3) → m_1
  105. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  106. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  107. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  108. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  109. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  110. Rotate(T, 15°, C) → T'
  111. Angle(T, C, T') → ε
  112. Ray(C, T') → q_1
  113. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  114. Vector(C, C'_8) → u_2
  115. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  116. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  117. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  118. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  119. Segment(B_2, C_1) → n_1
  120. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  121. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  122. Segment(D_1, E_1) → p_1
  123. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  124. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  125. Segment(F_1, N_1) → r_1
  126. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  127. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  128. Segment(O_1, P_1) → s_1
  129. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  130. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  131. Vector(B_2, Q_1) → c_1
  132. Translate(s, c_1) → s'_2
  133. Segment(C, K_1) → e_1
  134. Line(A, M_1) → t_1
  135. Intersect(t_1, e_1) → R_1
  136. Segment(A, R_1) → f_2
  137. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  138. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  139. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  140. Line(R_1, p_1) → g_3
  141. Point(g_3) → U_1
  142. Angle(U_1, R_1, K_1) → ζ
  143. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  144. Dilate(K_1, If[15 < e ≤ 16, 16 - e, 0], M_1) → W_1
  145. Segment(K_1, W_1) → p_2
  146. Line(A, K_1) → k_3
  147. Dilate(A, If[20 < e ≤ 21, 21 - e, 0], V_1) → A'_5
  148. Segment(A, A'_5) → s_3
  149. Ray(M_1, V_1) → j_3
  150. Circle(V_1, 1) → e_3
  151. Intersect(e_3, j_3, 2) → T_1
  152. Ray(V_1, T_1) → n_3
  153. Circle(V_1, (0.2 * r_2)) → p_3
  154. Intersect(p_3, n_3, 1) → A_2
  155. Intersect(p_3, f'', 2) → C_2
  156. Intersect(f_4, h_3) → H_2
  157. Dilate(C, If[17 < e ≤ 18, 18 - e, 0], H_2) → C'_7
  158. Segment(C, C'_7) → q_3
  159. Angle(A_2, V_1, C_2) → η
  160. Rotate(C_2, If[17 < e ≤ 18, ((-(e - 17)) * π / 4), ((-π)) / 4], V_1) → D_2
  161. CircleSector(V_1, D_2, C_2) → r_3
  162. Dilate(V_1, If[18 < e ≤ 19, 19 - e, 0], C) → E_2
  163. Vector(V_1, E_2) → v_2
  164. Translate(r_3, v_2) → t_3
  165. Rotate(t_3, If[19 < e ≤ 20, (((-(e - 19)) * 5) * π / 4), (-5 * π / 4)], C) → e_4
  166. Dilate(V_1, If[21 < e ≤ 22, 22 - e, 0], H_2) → F_2
  167. Segment(V_1, F_2) → g_4
  168. Rotate(A, If[22 < e ≤ 23, ((e - 22) * π / 2), π / 2], V_1) → A'_2
  169. Rotate(A'_5, If[22 < e ≤ 23, ((e - 22) * π / 2), π / 2], V_1) → W
  170. Segment(A'_2, W) → b_1
  171. Line(V_1, s_1) → j_2
  172. Dilate(B_2, If[23 < e ≤ 24, 24 - e, 0], V_1) → G_1
  173. Vector(B_2, G_1) → w_2
  174. Translate(s, w_2) → s'_3
  175. Intersect(t, f'', 1) → G_2
  176. Angle(G_2, V_1, H_2) → θ
  177. Rotate(s'_3, If[23 < e ≤ 24, ((e - 23) * θ), θ], G_1) → c_2
  178. Dilate(A, If[24 < e ≤ 25, 25 - e, 0], H_2) → A'_6
  179. Segment(A, A'_6) → l_2
  180. Dilate(V_1, If[25 < e ≤ 26, 26 - e, 0], A) → I_2
  181. Vector(V_1, I_2) → a_2
  182. Translate(r_3, a_2) → d_2
  183. Line(A, V_1) → b_2
  184. Line(V_1, l_2) → a_3
  185. Point(n_3) → J_2
  186. Point(a_3) → K_2
  187. Angle(K_2, V_1, J_2) → κ
  188. Rotate(d_2, If[26 < e ≤ 27, ((-(e - 26)) * κ), (-κ)], A) → d_3
  189. Dilate(L_1, If[17 < e ≤ 18, 18 - e, 0], H_2) → L_2
  190. Segment(L_1, L_2) → i_3
  191. Segment(K_1, H_2) → c_3

AI 绘图提示语参考

画两个动态的平方图形，展示它们之间的几何关系。包含旋转、反射和平移等变换，以及平行四边形、中点、垂线和辅助圆弧。通过滑块或可拖动元素，动态观察线段、角度和面积的关系，并连接到最大公约数和最小公倍数。图形应包含多个点、线、多边形和圆弧，以探索定理证明和基于变换的推理。
以上提示语为根据原始绘图命令整理而来，仅供参考。