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三个黄金矩形的惊喜

三个黄金矩形的惊喜
几何画板及绘图内容加载中，请稍等……
2017-09-03 11:45
变换几何作图
所属专题 GeoGebra 3D & AR: PreCalc & Calculus Resources
这个动态可修改的示例展示了三个黄金矩形的奇妙之处。黄金矩形是一种特殊的矩形，其长宽比为黄金比例。本示例通过几何作图展示了黄金矩形的性质，包括构造矩形、使用Dilate命令进行相似变换、以及通过旋转和距离计算来验证黄金比例。适合初中及以上学生学习几何变换和比例关系。

GeoGebra 命令及参数

  1. Line(F, H) → f
  2. Line(F, G) → h
  3. Line(H, E) → i
  4. Circle(F, H, xOyPlane) → c
  5. Intersect(h, c) → M, N
  6. OrthogonalLine(M, h, xOyPlane) → j
  7. Intersect(i, j) → O
  8. Midpoint(H, O) → P
  9. Distance(P, M) → distancePM
  10. Distance(M, E) → distanceME
  11. Dilate(F, If[0 < b ≤ 1, 1 - b, 0], H) → F'
  12. Segment(F, F') → k
  13. Rotate(F, If[1 < b ≤ 2, ((-(b - 1)) * π / 2), ((-π)) / 2], H) → F'_1
  14. Rotate(F', If[1 < b ≤ 2, ((-(b - 1)) * π / 2), ((-π)) / 2], H) → F''
  15. Segment(F'_1, F'') → k'
  16. Rotate(F'_1, If[2 < b ≤ 3, ((-(b - 2)) * π / 2), ((-π)) / 2], O) → Q
  17. Rotate(F'', If[2 < b ≤ 3, ((-(b - 2)) * π / 2), ((-π)) / 2], O) → F'''
  18. Segment(Q, F''') → k''
  19. Rotate(Q, If[3 < b ≤ 4, ((-(b - 3)) * π / 2), ((-π)) / 2], M) → Q'
  20. Rotate(F''', If[3 < b ≤ 4, ((-(b - 3)) * π / 2), ((-π)) / 2], M) → R
  21. Segment(Q', R) → k'''
  22. Angle(M, O, H, xOyPlane) → α
  23. Angle(O, H, F, xOyPlane) → β
  24. Angle(F, M, O, xOyPlane) → γ
  25. Angle(H, F, M, xOyPlane) → δ
  26. Polygon(F, H, O, M) → poly1, f_1, h_1, o, m
  27. Dilate(H, If[4 < b ≤ 5, 5 - b, 0], P) → H'
  28. Segment(H, H') → l
  29. Rotate(H, If[5 < b ≤ 6, ((-(b - 5)) * π), (-π)], P) → H'_1
  30. Rotate(H', If[5 < b ≤ 6, ((-(b - 5)) * π), (-π)], P) → H''
  31. Segment(H'_1, H'') → l'
  32. Dilate(P, If[6 < b ≤ 7, 7 - b, 0], M) → P'
  33. Segment(P, P') → n
  34. Angle(E, P, M, xOyPlane) → ε
  35. Rotate(P, If[7 < b ≤ 8, ((-(b - 7)) * ε), (-ε)], P) → P'_1
  36. Rotate(P', If[7 < b ≤ 8, ((-(b - 7)) * ε), (-ε)], P) → P''
  37. Segment(P'_1, P'') → n'
  38. CircleArc(P, P'', M, xOyPlane) → d
  39. Dilate(E, If[8 < b ≤ 9, 9 - b, 0], G) → E'
  40. Dilate(M, If[8 < b ≤ 9, 9 - b, 0], G) → M'
  41. Segment(E, E') → p
  42. Segment(M, M') → q
  43. Angle(F, G, E, xOyPlane) → ζ
  44. Polygon(M, O, E, G) → poly2, m_1, o_1, e, g_1
  45. Dilate(G, If[9 < b ≤ 10, 10 - b, 0], A) → G'
  46. Vector(G, G') → u
  47. Translate(E, u) → E'_1
  48. Translate(E', u) → E''
  49. Segment(E'_1, E'') → p'
  50. Plane(C, A, E'_1) → r
  51. Rotate(E'_1, If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], A, r) → S
  52. Rotate(E'', If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], A, r) → E'''
  53. Segment(S, E''') → p''
  54. Segment(F, G) → s
  55. Dilate(F, If[11 < b ≤ 12, 12 - b, 0], C) → F'_2
  56. Vector(F, F'_2) → v
  57. Translate(F, v) → F'_3
  58. Translate(G, v) → G'_1
  59. Segment(F'_3, G'_1) → s'
  60. Plane(T, F'_3, G'_1) → t
  61. Rotate(F'_3, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], F'_3, t) → U
  62. Rotate(G'_1, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], F'_3, t) → V
  63. Segment(U, V) → s''
  64. Angle(B, C, A) → η
  65. Dilate(C, If[12 < b ≤ 13, 13 - b, 0], A) → C'
  66. Vector(C, C') → w
  67. Translate(U, w) → U'
  68. Translate(V, w) → V'
  69. Segment(U', V') → s'''
  70. Angle(C, A, D) → θ
  71. Dilate(B, If[13 < b ≤ 14, 14 - b, 0], D) → B'
  72. Segment(B, B') → i_1
  73. Polygon(B, D, A, C) → poly3, b_1, d_1, a_1, c_1
  74. Dilate(A, If[14 < b ≤ 15, 15 - b, 0], I) → A'
  75. Vector(A, A') → u_1
  76. Translate(S, u_1) → S'
  77. Translate(E''', u_1) → W
  78. Segment(S', W) → p'''
  79. Plane(Z, I, J) → e_1
  80. Rotate(S', If[15 < b ≤ 16, ((b - 15) * π / 2), π / 2], I, e_1) → S''
  81. Rotate(W, If[15 < b ≤ 16, ((b - 15) * π / 2), π / 2], I, e_1) → W'
  82. Segment(S'', W') → j_1
  83. Dilate(B, If[16 < b ≤ 17, 17 - b, 0], J) → B'_1
  84. Vector(B, B'_1) → v_1
  85. Translate(U, v_1) → U'_1
  86. Translate(V, v_1) → V'_1
  87. Segment(U'_1, V'_1) → s'''_1
  88. Plane(A_1, B'_1, U'_1) → k_1
  89. Rotate(U'_1, If[15 < b ≤ 17, ((b - 16) * π / 2), π / 2], V'_1, k_1) → B_1
  90. Rotate(V'_1, If[15 < b ≤ 17, ((b - 16) * π / 2), π / 2], V'_1, k_1) → C_1
  91. Segment(B_1, C_1) → l_1
  92. Angle(I, J, L) → ι
  93. Dilate(L, If[17 < b ≤ 18, 18 - b, 0], K) → L'
  94. Vector(L, L') → w_1
  95. Translate(B_1, w_1) → D_1
  96. Translate(C_1, w_1) → E_1
  97. Segment(D_1, E_1) → n_1
  98. Angle(K, I, J) → κ
  99. Segment(L, D_1) → p_1
  100. Polygon(L, J, I, K) → poly4, l_2, j_2, i_2, k_2
  101. Dilate(F, If[20 < b ≤ 21, 21 - b, 0], H) → F'_4
  102. Segment(F, F'_4) → q_1
  103. Plane(F, C, H) → r_1
  104. Rotate(F, If[21 < b ≤ 22, ((-(b - 21)) * π / 3), ((-π)) / 3], F, r_1) → F'_5
  105. Rotate(F'_4, If[21 < b ≤ 22, ((-(b - 21)) * π / 3), ((-π)) / 3], F, r_1) → F_1
  106. Segment(F'_5, F_1) → s_1
  107. Plane(K, C, F) → t_1
  108. Rotate(F'_5, If[22 < b ≤ 23, ((b - 22) * π / 3), π / 3], C, t_1) → G_1
  109. Rotate(F_1, If[22 < b ≤ 23, ((b - 22) * π / 3), π / 3], C, t_1) → H_1
  110. Segment(G_1, H_1) → f_2
  111. Plane(C, K, A) → a_2
  112. Rotate(G_1, If[23 < b ≤ 24, ((-(b - 23)) * π / 3), ((-π)) / 3], K, a_2) → I_1
  113. Rotate(H_1, If[23 < b ≤ 24, ((-(b - 23)) * π / 3), ((-π)) / 3], K, a_2) → J_1
  114. Segment(I_1, J_1) → g_2
  115. Plane(A, K, G) → b_2
  116. Rotate(I_1, If[24 < b ≤ 25, ((b - 24) * π / 3), π / 3], A, b_2) → K_1
  117. Rotate(J_1, If[24 < b ≤ 25, ((b - 24) * π / 3), π / 3], A, b_2) → L_1
  118. Segment(K_1, L_1) → h_2
  119. Plane(A, G, E) → c_2
  120. Rotate(K_1, If[25 < b ≤ 26, ((-(b - 25)) * π / 3), ((-π)) / 3], G, c_2) → M_1
  121. Rotate(L_1, If[25 < b ≤ 26, ((-(b - 25)) * π / 3), ((-π)) / 3], G, c_2) → N_1
  122. Segment(M_1, N_1) → m_2
  123. Plane(G, E, D) → d_2
  124. Rotate(M_1, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], E, d_2) → O_1
  125. Rotate(N_1, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], E, d_2) → P_1
  126. Segment(O_1, P_1) → n_2
  127. Plane(F, H, B) → e_2
  128. Rotate(F, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], H, e_2) → F'_6
  129. Rotate(F'_4, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], H, e_2) → Q_1
  130. Segment(F'_6, Q_1) → p_2
  131. Plane(E, D, J) → o_2
  132. Rotate(O_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], D, o_2) → R_1
  133. Rotate(P_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], D, o_2) → S_1
  134. Segment(R_1, S_1) → q_2
  135. Point(s_1) → T_1
  136. Plane(H, B, J) → r_2
  137. Rotate(F'_6, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], B, r_2) → U_1
  138. Rotate(Q_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], B, r_2) → V_1
  139. Segment(U_1, V_1) → s_2
  140. Rotate(U_1, If[28 < b ≤ 29, ((-(b - 28)) * π / 3), ((-π)) / 3], J, r_2) → W_1
  141. Rotate(V_1, If[28 < b ≤ 29, ((-(b - 28)) * π / 3), ((-π)) / 3], J, r_2) → Z_1
  142. Segment(W_1, Z_1) → t_2
  143. Rotate(M_1, If[28 < b ≤ 29, ((b - 28) * π / 3), π / 3], G, d_2) → A_2
  144. Rotate(N_1, If[28 < b ≤ 29, ((b - 28) * π / 3), π / 3], G, d_2) → B_2
  145. Segment(A_2, B_2) → f_3
  146. Intersect(q_1, t_2) → C_2
  147. Plane(H, C, A) → u_2
  148. Plane(F, B, D) → v_2
  149. Rotate(W_1, If[29 < b ≤ 30, (((-(b - 29)) * 108) * π / 180), (-108 * π / 180)], H, u_2) → D_2
  150. Rotate(Z_1, If[29 < b ≤ 30, (((-(b - 29)) * 108) * π / 180), (-108 * π / 180)], H, u_2) → E_2
  151. Segment(D_2, E_2) → g_3
  152. Rotate(D_2, If[30 < b ≤ 31, (((-(b - 30)) * 108) * π / 180), (-108 * π / 180)], C, u_2) → F_2
  153. Rotate(E_2, If[30 < b ≤ 31, (((-(b - 30)) * 108) * π / 180), (-108 * π / 180)], C, u_2) → G_2
  154. Segment(F_2, G_2) → h_3
  155. Rotate(F_2, If[31 < b ≤ 32, (((-(b - 31)) * 108) * π / 180), (-108 * π / 180)], A, u_2) → H_2
  156. Rotate(G_2, If[31 < b ≤ 32, (((-(b - 31)) * 108) * π / 180), (-108 * π / 180)], A, u_2) → I_2
  157. Segment(H_2, I_2) → i_3
  158. Rotate(H_2, If[32 < b ≤ 33, (((-(b - 32)) * 108) * π / 180), (-108 * π / 180)], E, u_2) → J_2
  159. Rotate(I_2, If[32 < b ≤ 33, (((-(b - 32)) * 108) * π / 180), (-108 * π / 180)], E, u_2) → K_2
  160. Segment(J_2, K_2) → j_3
  161. Rotate(A_2, If[29 < b ≤ 30, (((b - 29) * 108) * π / 180), (108 * π / 180)], D, v_2) → L_2
  162. Rotate(B_2, If[30 < b ≤ 31, (((b - 30) * 108) * π / 180), (108 * π / 180)], D, v_2) → M_2
  163. Segment(L_2, M_2) → k_3
  164. Rotate(L_2, If[31 < b ≤ 32, (((b - 31) * 108) * π / 180), (108 * π / 180)], B, v_2) → N_2
  165. Rotate(M_2, If[30 < b ≤ 31, (((b - 30) * 108) * π / 180), (108 * π / 180)], B, v_2) → O_2
  166. Segment(N_2, O_2) → l_3
  167. Rotate(N_2, If[31 < b ≤ 32, (((b - 31) * 108) * π / 180), (108 * π / 180)], F, v_2) → P_2
  168. Rotate(O_2, If[32 < b ≤ 33, (((b - 32) * 108) * π / 180), (108 * π / 180)], F, v_2) → Q_2
  169. Segment(P_2, Q_2) → m_3
  170. Rotate(P_2, If[33 < b ≤ 34, (((b - 33) * 108) * π / 180), (108 * π / 180)], K, v_2) → R_2
  171. Rotate(Q_2, If[32 < b ≤ 33, (((b - 32) * 108) * π / 180), (108 * π / 180)], K, v_2) → S_2
  172. Segment(R_2, S_2) → n_3
  173. Plane(E, I, J) → w_2
  174. Rotate(J_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], J, w_2) → T_2
  175. Rotate(K_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], J, w_2) → U_2
  176. Segment(T_2, U_2) → p_3
  177. Rotate(H_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], E, Plane[A, E, I]) → V_2
  178. Rotate(I_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], E, Plane[A, E, I]) → W_2
  179. Segment(V_2, W_2) → q_3
  180. Rotate(F_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], A, Plane[A, C, I]) → Z_2
  181. Rotate(G_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], A, Plane[A, C, I]) → A_3
  182. Segment(Z_2, A_3) → r_3
  183. Rotate(D_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], C, Plane[H, C, I]) → B_3
  184. Rotate(E_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], C, Plane[H, C, I]) → C_3
  185. Segment(B_3, C_3) → s_3
  186. Rotate(W_1, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], H, Plane[H, J, I]) → D_3
  187. Rotate(Z_1, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], H, Plane[H, J, I]) → E_3
  188. Segment(D_3, E_3) → t_3
  189. Rotate(P_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], F, Plane[K, F, L]) → F_3
  190. Rotate(Q_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], F, Plane[K, F, L]) → G_3
  191. Segment(F_3, G_3) → a_3
  192. Rotate(R_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], K, Plane[K, G, L]) → H_3
  193. Rotate(S_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], K, Plane[K, G, L]) → I_3
  194. Segment(H_3, I_3) → b_3
  195. Rotate(A_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], G, Plane[D, G, L]) → J_3
  196. Rotate(B_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], G, Plane[D, G, L]) → K_3
  197. Segment(J_3, K_3) → c_3
  198. Rotate(L_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], D, Plane[D, B, L]) → L_3
  199. Rotate(M_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], D, Plane[D, B, L]) → M_3
  200. Segment(L_3, M_3) → d_3
  201. Rotate(N_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], B, Plane[F, B, L]) → N_3
  202. Rotate(O_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], B, Plane[F, B, L]) → O_3
  203. Segment(N_3, O_3) → e_3
  204. Polygon(C, I, A) → poly5, a_4, c_4, i_4
  205. Polygon(I, E, A) → poly6, a_5, i_5, e_4
  206. Polygon(E, G, A) → poly7, a_6, e_5, g_4
  207. Polygon(G, K, A) → poly8, a_7, g_5, k_4
  208. Polygon(K, C, A) → poly9, a_8, k_5, c_5
  209. Polygon(K, F, C) → poly10, c_6, k_6, f_4
  210. Polygon(F, H, C) → poly11, c_7, f_5, h_4
  211. Polygon(H, I, C) → poly12, c_8, h_5, i_6
  212. Polygon(H, J, I) → poly13, i_7, h_6, j_4
  213. Polygon(J, E, I) → poly14, i_8, j_5, e_6
  214. Polygon(H, B, J) → poly15, j_6, h_7, b_4
  215. Polygon(H, B, F) → poly16, f_6, h_8, b_5
  216. Polygon(B, F, L) → poly17, l_4, b_6, f_7
  217. Polygon(F, K, L) → poly18, l_5, f_8, k_7
  218. Polygon(L, K, G) → poly19, g_6, l_6, k_8
  219. Polygon(L, G, D) → poly20, d_4, l_7, g_7
  220. Polygon(B, L, D) → poly21, d_5, b_7, l_8
  221. Polygon(D, E, J) → poly22, j_7, d_6, e_7
  222. Polygon(G, D, E) → poly23, e_8, g_8, d_7
  223. Polygon(B, D, J) → poly24, j_8, b_8, d_8
  224. Line((0, 0, 0), (1, 0, 0)) → m_4
  225. Line((0, 0, 0), (0, 1, 0)) → n_4
  226. Line((0, 0, 0), (0, 0, 1)) → p_4