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几何画板动态演示：菱形的旋转现象

几何画板动态演示：菱形的旋转现象
几何画板及绘图内容加载中，请稍等……
2018-09-15 08:23
多边形几何作图
这是一个动态演示的几何挑战证明问题，展示了菱形在旋转过程中的有趣现象。涉及的核心数学概念包括菱形的性质和旋转变换。通过使用GeoGebra命令如Segment, Rotate, Angle, Polygon等进行构造。适合初中及以上学生学习几何变换和证明。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Polygon(A, B, C) → t1, c, a_1, b
  7. Segment(A, C) → h
  8. Segment(C, B) → i
  9. Circle(A, f / 3) → d
  10. Intersect(d, f, 1) → D
  11. Midpoint(D, B) → E
  12. Circle(A, h / 3) → e
  13. Intersect(e, h, 1) → F
  14. Midpoint(F, C) → G
  15. Circle(C, i / 3) → k
  16. Intersect(k, i, 1) → H
  17. Midpoint(H, B) → I
  18. Segment(A, H) → j
  19. Segment(A, I) → l
  20. Segment(C, D) → m
  21. Segment(C, E) → n
  22. Segment(B, F) → p
  23. Segment(B, G) → q
  24. Intersect(j, q) → J
  25. Intersect(m, p) → K
  26. Intersect(l, n) → L
  27. Polygon(K, L, J) → t2, j_1, k_1, l_1
  28. Vector(K, A) → u
  29. Translate(K, u) → K'
  30. Translate(L, u) → L'
  31. Translate(J, u) → J'
  32. Polygon(K', L', J') → t2', j', k', l'
  33. Vector(K, L') → v
  34. Translate(K, v) → K'_1
  35. Translate(L, v) → L'_1
  36. Translate(J, v) → J'_1
  37. Polygon(K'_1, L'_1, J'_1) → t2'_1, j'_1, k'_1, l'_1
  38. Vector(L, E) → w
  39. Vector(J, C) → r
  40. Translate(K, r) → K'_2
  41. Translate(L, r) → L'_2
  42. Translate(J, r) → J'_2
  43. Polygon(K'_2, L'_2, J'_2) → t2'_2, j'_2, k'_2, l'_2
  44. Rotate(K, π, J) → K'_3
  45. Rotate(L, π, J) → L'_3
  46. Rotate(J, π, J) → J'_3
  47. Polygon(K'_3, L'_3, J'_3) → t2'_3, j'_3, k'_3, l'_3
  48. Vector(K, J') → t
  49. Translate(K, t) → K'_4
  50. Translate(L, t) → L'_4
  51. Translate(J, t) → J'_4
  52. Polygon(K'_4, L'_4, J'_4) → t2'_4, j'_4, k'_4, l'_4
  53. Circle(L'_1, j') → c_1
  54. Intersect(c_1, f, 2) → M
  55. Vector(L, M) → u_1
  56. Translate(K, u_1) → K'_5
  57. Translate(L, u_1) → L'_5
  58. Translate(J, u_1) → J'_5
  59. Polygon(K'_5, L'_5, J'_5) → t2'_5, j'_5, k'_5, l'_5
  60. Vector(L', L'_1) → v_1
  61. Translate(K'_5, v_1) → N
  62. Translate(L'_5, v_1) → O
  63. Translate(J'_5, v_1) → P
  64. Polygon(N, O, P) → t3, p_1, n_1, o_1
  65. Translate(K, v_1) → K'_6
  66. Translate(L, v_1) → L'_6
  67. Translate(J, v_1) → J'_6
  68. Polygon(K'_6, L'_6, J'_6) → t2'_6, j'_6, k'_6, l'_6
  69. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'
  70. Segment(A, A') → f_1
  71. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], D) → A'_1
  72. Segment(D, A'_1) → g_1
  73. Rotate(D, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], E) → D'
  74. Segment(E, D') → h_1
  75. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], I) → B'_1
  76. Segment(B, B'_1) → i_1
  77. Rotate(B, If[4 < o ≤ 5, ((-(o - 4)) * π), (-π)], I) → B'_2
  78. Segment(I, B'_2) → m_1
  79. Rotate(I, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], H) → I'
  80. Segment(H, I') → q_1
  81. Dilate(C, If[6 < o ≤ 7, 7 - o, 0], G) → C'
  82. Segment(C, C') → r_1
  83. Rotate(C, If[7 < o ≤ 8, ((-(o - 7)) * π), (-π)], G) → C'_1
  84. Segment(G, C'_1) → s_1
  85. Rotate(G, If[8 < o ≤ 9, ((-(o - 8)) * π), (-π)], F) → G'
  86. Segment(F, G') → t_1
  87. Dilate(A, If[10 < o ≤ 11, 11 - o, 0], J) → A'_2
  88. Segment(A, A'_2) → b_1
  89. Dilate(B, If[10 < o ≤ 11, 11 - o, 0], J) → B'_3
  90. Segment(B, B'_3) → d_1
  91. Dilate(J, If[11 < o ≤ 12, 12 - o, 0], G) → J'_7
  92. Dilate(J, If[11 < o ≤ 12, 12 - o, 0], H) → J'_8
  93. Segment(J, J'_7) → e_1
  94. Segment(J, J'_8) → f_2
  95. Dilate(A, If[12 < o ≤ 13, 13 - o, 0], L) → A'_3
  96. Dilate(C, If[12 < o ≤ 13, 13 - o, 0], L) → C'_2
  97. Segment(C, C'_2) → g_2
  98. Segment(A, A'_3) → h_2
  99. Dilate(L, If[13 < o ≤ 14, 14 - o, 0], E) → L'_7
  100. Dilate(L, If[13 < o ≤ 14, 14 - o, 0], I) → L'_8
  101. Segment(L, L'_8) → i_2
  102. Segment(L, L'_7) → j_2
  103. Dilate(C, If[14 < o ≤ 15, 15 - o, 0], K) → C'_3
  104. Dilate(B, If[14 < o ≤ 15, 15 - o, 0], K) → B'_4
  105. Segment(C, C'_3) → k_2
  106. Segment(B, B'_4) → l_2
  107. Dilate(K, If[15 < o ≤ 16, 16 - o, 0], D) → K'_7
  108. Dilate(K, If[15 < o ≤ 16, 16 - o, 0], F) → K'_8
  109. Segment(K, K'_7) → m_2
  110. Segment(K, K'_8) → n_2
  111. Dilate(J, If[18 < o ≤ 19, 19 - o, 0], K) → J'_9
  112. Segment(J, J'_9) → p_2
  113. Dilate(K, If[18 < o ≤ 19, 19 - o, 0], L) → K'_9
  114. Segment(K, K'_9) → q_2
  115. Dilate(L, If[18 < o ≤ 19, 19 - o, 0], J) → L'_9
  116. Segment(L, L'_9) → r_2
  117. Dilate(K, If[23 < o ≤ 24, 24 - o, 0], A) → K'_{10}
  118. Vector(K, K'_{10}) → w_1
  119. Translate(K, w_1) → K'_{11}
  120. Translate(L, w_1) → L'_{10}
  121. Translate(J, w_1) → J'_{10}
  122. Polygon(K'_{11}, L'_{10}, J'_{10}) → t2'_7, j'_{10}, k'_{11}, l'_{10}
  123. Translate(J, w_1) → J'_{11}
  124. Translate(L, w_1) → L'_{11}
  125. Midpoint(A, B) → R
  126. Midpoint(A, C) → S
  127. Midpoint(C, B) → T
  128. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], K'_1) → A'_4
  129. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], R) → A'_5
  130. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], N) → A'_6
  131. Vector(A, A'_4) → u_2
  132. Vector(A, A'_5) → v_2
  133. Vector(A, A'_6) → w_2
  134. Translate(K'_{11}, u_2) → Q
  135. Translate(L'_{10}, u_2) → U
  136. Translate(J'_{10}, u_2) → V
  137. Polygon(Q, U, V) → t4, v_3, q_3, u_3
  138. Translate(K'_{11}, w_2) → W
  139. Translate(L'_{10}, w_2) → Z
  140. Translate(J'_{10}, w_2) → A_1
  141. Polygon(W, Z, A_1) → t5, a_2, w_3, z_1
  142. Translate(K'_{11}, v_2) → B_1
  143. Translate(L'_{10}, v_2) → C_1
  144. Translate(J'_{10}, v_2) → D_1
  145. Polygon(B_1, C_1, D_1) → t6, d_2, b_2, c_2
  146. Dilate(K'_1, If[26 < o ≤ 27, 27 - o, 0], K) → E_1
  147. Vector(K'_1, E_1) → e_2
  148. Translate(Q, e_2) → Q'
  149. Translate(U, e_2) → U'
  150. Translate(V, e_2) → V'
  151. Polygon(Q', U', V') → t4', v', q', u'
  152. Translate(B_1, e_2) → F_1
  153. Translate(C_1, e_2) → G_1
  154. Translate(D_1, e_2) → H_1
  155. Polygon(F_1, G_1, H_1) → t6', h_3, f_3, g_3
  156. Translate(K'_{11}, Vector[(2 * e_2)]) → I_1
  157. Translate(L'_{10}, Vector[(2 * e_2)]) → J_1
  158. Translate(J'_{10}, Vector[(2 * e_2)]) → K_1
  159. Polygon(I_1, J_1, K_1) → t7, k_3, i_3, j_3
  160. Translate(K'_{11}, e_2) → L_1
  161. Translate(L'_{10}, e_2) → M_1
  162. Translate(J'_{10}, e_2) → N_1
  163. Polygon(L_1, M_1, N_1) → t8, n_3, l_3, m_3
  164. Translate(K'_{11}, Vector[(3 * e_2)]) → O_1
  165. Translate(L'_{10}, Vector[(3 * e_2)]) → P_1
  166. Translate(J'_{10}, Vector[(3 * e_2)]) → Q_1
  167. Polygon(O_1, P_1, Q_1) → t9, q_4, o_3, p_3
  168. Translate(Q, Vector[(2 * e_2)]) → Q'_1
  169. Translate(U, Vector[(2 * e_2)]) → U'_1
  170. Translate(V, Vector[(2 * e_2)]) → V'_1
  171. Polygon(Q'_1, U'_1, V'_1) → t4'_1, v'_1, q'_1, u'_1
  172. Midpoint(r_2) → R_1
  173. Rotate(Q', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → Q''
  174. Rotate(U', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → U''
  175. Rotate(V', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → V''
  176. Polygon(Q'', U'', V'') → t4'', v'', q'', u''
  177. Vector(L, N) → s_2
  178. Translate(Q'', s_2) → Q'''
  179. Translate(U'', s_2) → U'''
  180. Translate(V'', s_2) → V'''
  181. Polygon(Q''', U''', V''') → t4''', v''', q''', u'''
  182. Translate(Q'', Vector[(-s_2)]) → Q'''_1
  183. Translate(U'', Vector[(-s_2)]) → U'''_1
  184. Translate(V'', Vector[(-s_2)]) → V'''_1
  185. Polygon(Q'''_1, U'''_1, V'''_1) → t4'''_1, v'''_1, q'''_1, u'''_1
  186. Vector(L, K) → t_2
  187. Translate(Q'', t_2) → Q'''_2
  188. Translate(U'', t_2) → U'''_2
  189. Translate(V'', t_2) → V'''_2
  190. Polygon(Q'''_2, U'''_2, V'''_2) → t4'''_2, v'''_2, q'''_2, u'''_2
  191. Translate(Q''', t_2) → S_1
  192. Translate(U''', t_2) → T_1
  193. Translate(V''', t_2) → U_1
  194. Polygon(S_1, T_1, U_1) → t10, u_4, s_3, t_3
  195. Translate(S_1, t_2) → V_1
  196. Translate(T_1, t_2) → W_1
  197. Translate(U_1, t_2) → Z_1
  198. Polygon(V_1, W_1, Z_1) → t10', z_2, v_4, w_4
  199. Polygon(J, K, L) → t11, l_4, j_4, k_4

AI 绘图提示语参考

绘制一个菱形，并围绕一个顶点旋转它。首先，创建一个菱形，然后围绕一个顶点旋转这个菱形，观察旋转过程中菱形的动态变化。具体步骤包括创建线段、旋转点、构造角度、绘制多边形和圆，最后通过平移和旋转操作展示菱形的旋转现象。
以上提示语为根据原始绘图命令整理而来，仅供参考。