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Finsler-Hadwiger 定理续篇：正方形与线段的动态几何演示

Finsler-Hadwiger 定理续篇：正方形与线段的动态几何演示
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2017-04-25 01:10
多边形向量变换几何作图
① 本范例演示 Finsler-Hadwiger 定理的后续构造，展示两个正方形通过旋转、缩放、平移等变换形成的几何关系。② 涉及核心数学概念包括：正方形的性质、线段中点、向量平移、旋转变换、相似与全等、几何变换群。③ 作图要点：使用 Circle、Point、Polygon、Midpoint、Dilate、Segment、Rotate、Translate、Line、Angle、OrthogonalLine、Ray 等命令，构造点 A、B、C、D 等顶点、多边形 poly1、poly2 及其变换后的对象 B'、B'_1、B'_2 等。④ 适合高中几何选修、大学几何变换课程及 GeoGebra 进阶教学演示。

GeoGebra 命令及参数

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(D, A) → H
  8. Midpoint(F, C) → I
  9. Midpoint(D, C) → J
  10. Midpoint(A, F) → K
  11. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  12. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  13. Segment(B, B') → n
  14. Segment(B, B'_1) → p
  15. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  16. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  17. Segment(B'_2, L) → p'
  18. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  19. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  20. Segment(B'_3, B'') → n'
  21. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  22. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  23. Segment(M, L') → p''
  24. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  25. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  26. Segment(N, B''') → n''
  27. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  28. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  29. Segment(N', O) → n'''
  30. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  31. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  32. Segment(M', L'') → p'''
  33. Segment(F, G) → q
  34. Segment(F, B) → r
  35. Segment(A, B) → s
  36. Min(r, s) → o
  37. Circle(F, (0.15 * o)) → t
  38. Intersect(l, q) → P
  39. Intersect(t, q) → Q,
  40. Intersect(t, r) → R,
  41. CircleSector(F, Q, R) → c_1
  42. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  43. Vector(F, F') → u
  44. Translate(c_1, u) → d_1
  45. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  46. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  47. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  48. Vector(F, F'_1) → v
  49. Vector(C, C') → w
  50. Translate(e_1, w) → f_1
  51. Translate(c_1, v) → g_1
  52. Line(A, p) → h_1
  53. Line(E, p''') → i_1
  54. Point(h_1) → S
  55. Angle(S, A, E) → α
  56. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  57. Line(E, h_1) → j_1
  58. Point(j_1) → T
  59. Angle(D, E, T) → β
  60. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  61. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  62. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  63. Vector(E, E') → u_1
  64. Vector(F, F'_2) → v_1
  65. Translate(p_1, u_1) → q_1
  66. Translate(c_1, v_1) → r_1
  67. Line(G, n'') → l_1
  68. Point(l_1) → U
  69. Line(G, n') → m_1
  70. Point(m_1) → V
  71. Angle(C, G, V) → γ
  72. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  73. Line(D, p) → n_1
  74. Point(n_1) → W
  75. Angle(W, D, B) → δ
  76. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  77. Midpoint(A, B) → Z
  78. Midpoint(B, C) → A_1
  79. Midpoint(Z, B) → B_1
  80. Midpoint(A_1, B) → C_1
  81. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  82. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  83. Vector(A, A') → w_1
  84. Vector(C, C'_1) → a_1
  85. Translate(e_1, a_1) → c_2
  86. Translate(k_1, w_1) → d_2
  87. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  88. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  89. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  90. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  91. Vector(B_1, E_1) → b_1
  92. Vector(C_1, D_1) → u_2
  93. Translate(e_2, b_1) → g_2
  94. Translate(f_2, u_2) → h_2
  95. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  96. Segment(F, F'_3) → i_2
  97. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  98. Segment(D, D') → j_2
  99. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  100. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  101. Segment(F'_4, F_1) → k_2
  102. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  103. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  104. Segment(D'_1, D'') → l_2
  105. Dilate(A, If[14 < a ≤ 15, 15 - a, 0], K) → A'_1
  106. Dilate(C, If[14 < a ≤ 15, 15 - a, 0], J) → C'_2
  107. Segment(A, A'_1) → m_2
  108. Segment(C, C'_2) → n_2
  109. Dilate(K, If[15 < a ≤ 16, 16 - a, 0], F) → K'
  110. Dilate(J, If[15 < a ≤ 16, 16 - a, 0], D) → J'
  111. Segment(K, K') → p_2
  112. Segment(J, J') → q_2
  113. Rotate(A, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], K) → A'_2
  114. Rotate(A'_1, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], K) → G_1
  115. Segment(A'_2, G_1) → r_2
  116. Rotate(C, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], J) → C'_3
  117. Rotate(C'_2, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], J) → H_1
  118. Segment(C'_3, H_1) → s_2
  119. Dilate(K, If[17 < a ≤ 18, 18 - a, 0], I) → K'_1
  120. Rotate(K, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], K) → K'_2
  121. Rotate(K'_1, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], K) → I_1
  122. Rotate(K'_2, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], H) → J_1
  123. Rotate(I_1, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], H) → K_1
  124. Rotate(J_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], J) → L_1
  125. Rotate(K_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], J) → M_1
  126. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  127. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  128. Vector(B, B'_4) → v_2
  129. Vector(B, B'_5) → w_2
  130. Line(I, p) → g_3
  131. Point(g_3) → N_1
  132. Angle(N_1, I, J) → ε
  133. Line(H, n''') → l_3
  134. Point(l_3) → O_1
  135. Angle(K, H, O_1) → ζ
  136. Dilate(F, If[23 < a ≤ 24, 24 - a, 0], K) → F'_5
  137. Dilate(D, If[23 < a ≤ 24, 24 - a, 0], J) → D'_2
  138. Vector(D, D'_2) → u_3
  139. Vector(F, F'_5) → v_3
  140. Translate(t_1, u_3) → r_3
  141. Line(K, p) → n_3
  142. Point(n_3) → Q_1
  143. Angle(Q_1, K, I) → θ_1
  144. Line(A, K) → a_3
  145. Line(J, B) → b_3
  146. Polygon(A, F, B) → poly3, b_2, a_2, f_3
  147. Dilate(J, If[17 < a ≤ 18, 18 - a, 0], B) → J'_1
  148. Segment(J, J'_1) → t_2
  149. OrthogonalLine(J, r_2) → h_3
  150. Intersect(a_3, h_3) → P_1
  151. Dilate(B, If[18 < a ≤ 19, 19 - a, 0], P_1) → B'_6
  152. Segment(B, B'_6) → i_3
  153. Dilate(B, If[19 < a ≤ 20, 20 - a, 0], P_1) → B'_7
  154. Vector(B, B'_7) → w_3
  155. Translate(h_2, w_3) → k_3
  156. Line(B, r_2) → j_3
  157. Point(j_3) → R_1
  158. Angle(F, B, R_1) → θ
  159. Rotate(k_3, If[19 < a ≤ 20, ((a - 19) * θ), θ], B'_7) → s_3
  160. Line(B, m_2) → m_3
  161. Rotate(J, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], B) → J'_2
  162. Rotate(J'_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], B) → S_1
  163. Segment(J'_2, S_1) → f_4
  164. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], K) → B'_8
  165. Vector(B, B'_8) → c_3
  166. Translate(J'_2, c_3) → T_1
  167. Translate(S_1, c_3) → U_1
  168. Segment(T_1, U_1) → p_3
  169. Rotate(J, If[22 < a ≤ 23, ((-(a - 22)) * π / 2), ((-π)) / 2], B) → J'_3
  170. Rotate(J'_1, If[22 < a ≤ 23, ((-(a - 22)) * π / 2), ((-π)) / 2], B) → V_1
  171. Segment(J'_3, V_1) → d_3
  172. Dilate(B, If[23 < a ≤ 24, 24 - a, 0], K) → B'_9
  173. Vector(B, B'_9) → e_3
  174. Translate(J'_3, e_3) → W_1
  175. Translate(V_1, e_3) → Z_1
  176. Segment(W_1, Z_1) → q_3
  177. Line(A, K) → t_3
  178. Circle(A, 2) → c_4
  179. Intersect(c_4, t_3, 1) → A_2
  180. Circle(F, 2) → d_4
  181. Intersect(d_4, t_3, 2) → B_2
  182. Ray(A, A_2) → g_4
  183. Ray(F, B_2) → h_4

AI 绘图提示语参考

绘制两个正方形ABCD和CBFG，它们共享顶点B，其中C位于以B为圆心、半径小于AB的圆上。构造正方形后，取各边中点并连接关键线段。然后引入一个动画参数，按顺序演示以下变换过程：首先对顶点B进行缩放得到对应点并连线；接着以不同中心旋转得到新点对和对应线段；然后平移包含扇形区域的图形；最后再次旋转和缩放，展示变换前后图形的对应关系与角度标记。整个构造体现Finsler-Hadwiger定理的动态几何演绎。
以上提示语为根据原始绘图命令整理而来，仅供参考。