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字母 X 的对称轴：构造与验证

字母 X 的对称轴：构造与验证
几何画板及绘图内容加载中，请稍等……
2020-12-24 17:16
圆多边形角变换几何作图
该范例演示如何在几何环境中确定并可视化字母 X（由若干线段与曲线拼接）所具有的对称轴，并通过镜像与角度刻画展示对称性的对应关系。核心数学概念包括：对称轴与镜像变换（Mirror）、中点与线段平分线（Midpoint、LineBisector）、角度构造与测量（Angle）、以及圆与扇形/弧的几何辅助形状（Circle、CircleSector）来组合出最终图形。作图要点：利用 Polygon、Segment、Intersect 等命令生成组成元素并求交点；用 Circle 与 CircleSector 提供曲线或圆弧部分；借助 Point、Line 创建参考点与直线；再用 Mirror、Rotate、Translate 将关键结构进行镜像/旋转/平移以验证对称；同时通过 Midpoint、LineBisector 找到对称相关的基准线；最后用 Textfield 和 Text 标注关键结论（如对称轴位置）。适合中学至大学基础几何与数学建模的课堂演示，也适合用于对称性、变换几何与几何作图教学与自学。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → mnogokut1, f, g, h, i, C, D
  2. Segment(B, D) → k
  3. Segment(A, C) → j
  4. Intersect(j, k) → E
  5. Circle(E, B) → c
  6. Point(c) → F
  7. Line(F, E) → l
  8. Intersect(c, l, 2) → G
  9. CircleSector(E, G, F) → polukrug
  10. Point(l) → H
  11. Point(l) → I
  12. Segment(H, I) → m
  13. Polygon(J, J', 4) → mnogokut2, n, p, q, r, K, L
  14. Segment(J, K) → s
  15. Segment(J', L) → t
  16. Intersect(s, t) → M
  17. Line(M, l) → a
  18. Point(a) → N
  19. Point(a) → O
  20. Segment(N, O) → b
  21. Polygon(A, B, T_1) → mnogokut3, t_1, a_1, b_1
  22. Mirror(A, l) → A'
  23. Mirror(B, l) → B'
  24. Mirror(T_1, l) → P
  25. Polygon(A', B', P) → mnogokut3', p_1, a', b'
  26. Polygon(J, J', T_2) → mnogokut3_1, t_2, j_1, j'
  27. Midpoint(Segment[D, C]) → T_3
  28. Polygon(A, B, T_3) → mnogokut4, t_3, a_2, b_2
  29. Midpoint(Segment[L, K]) → T_4
  30. Polygon(J, J', T_4) → mnogokut4_1, t_4, j_2, j'_1
  31. Mirror(A, l) → A'_1
  32. Mirror(B, l) → B'_1
  33. Mirror(T_3, l) → Q
  34. Polygon(A'_1, B'_1, Q) → mnogokut4', q_1, a'_1, b'_1
  35. Polygon(A, B, 3) → mnogokut5, d, e, f_1, R
  36. LineBisector(d) → g_1
  37. LineBisector(e) → h_1
  38. Intersect(g_1, h_1) → S
  39. Circle(S, B) → c_1
  40. Point(c_1) → T
  41. Line(T, S) → i_1
  42. Intersect(c_1, i_1, 2) → U
  43. CircleSector(S, U, T) → d_1
  44. Point(i_1) → V
  45. Point(i_1) → W
  46. Segment(V, W) → k_1
  47. Mirror(mnogokut5, i_1) → mnogokut5'
  48. Polygon(J, J', 3) → mnogokut6, l_1, m_1, r_1, Z
  49. LineBisector(l_1) → s_1
  50. LineBisector(m_1) → e_1
  51. Intersect(s_1, e_1) → A_1
  52. Line(A_1, k_1) → f_2
  53. Point(f_2) → B_1
  54. Point(f_2) → C_1
  55. Segment(C_1, B_1) → g_2
  56. Polygon(A, B, 4) → mnogokut7, h_2, i_2, k_2, l_2, D_1, E_1
  57. Mirror(mnogokut7, l) → mnogokut7'
  58. Polygon(J, J', 4) → mnogokut7_1, m_2, n_2, p_2, q_2, F_1, G_1
  59. Line(A, I_1) → r_2
  60. Circle(A, Segment[A, H_1]) → c_2
  61. Intersect(c_2, r_2, 2) → J_1
  62. Circle(J_1, Segment[A, H_1]) → d_2
  63. Circle(H_1, Segment[A, H_1]) → e_2
  64. Intersect(d_2, e_2, 1) → K_1
  65. Polygon(A, H_1, K_1, J_1) → mnogokut8, a_3, h_3, k_3, j_3
  66. Segment(A, K_1) → s_2
  67. Segment(H_1, J_1) → f_3
  68. Intersect(s_2, f_3) → L_1
  69. Circle(L_1, K_1) → c_3
  70. Point(c_3) → M_1
  71. Line(M_1, L_1) → g_3
  72. Intersect(c_3, g_3, 2) → N_1
  73. CircleSector(L_1, N_1, M_1) → d_3
  74. Mirror(A, g_3) → A'_2
  75. Mirror(H_1, g_3) → O_1
  76. Mirror(K_1, g_3) → P_1
  77. Mirror(J_1, g_3) → Q_1
  78. Polygon(A'_2, O_1, P_1, Q_1) → mnogokut8', a'_2, o_1, p_3, q_3
  79. Point(g_3) → R_1
  80. Point(g_3) → S_1
  81. Segment(S_1, R_1) → i_3
  82. Line(J, r_2) → l_3
  83. Circle(J, Segment[J, U_1]) → e_3
  84. Intersect(e_3, l_3, 2) → V_1
  85. Circle(V_1, Segment[J, U_1]) → r_3
  86. Circle(U_1, Segment[J, U_1]) → s_3
  87. Intersect(r_3, s_3, 1) → W_1
  88. Polygon(J, U_1, W_1, V_1) → mnogokut9, j_4, u_1, w_1, v_1
  89. Segment(J, W_1) → m_3
  90. Segment(U_1, V_1) → n_3
  91. Intersect(m_3, n_3) → Z_1
  92. Line(Z_1, g_3) → b_3
  93. Point(b_3) → A_2
  94. Point(b_3) → B_2
  95. Segment(B_2, A_2) → f_4
  96. Circle(A, Segment[B, C_2]) → c_4
  97. Circle(C_2, Segment[A, B]) → d_4
  98. Intersect(c_4, d_4, 1) → D_2
  99. Polygon(A, B, C_2, D_2) → mnogokut10, a_4, b_4, c_5, d_5
  100. Segment(A, C_2) → g_4
  101. Segment(B, D_2) → h_4
  102. Intersect(g_4, h_4) → E_2
  103. Circle(E_2, B) → e_4
  104. Point(e_4) → F_2
  105. Line(F_2, E_2) → i_4
  106. Intersect(e_4, i_4, 2) → G_2
  107. CircleSector(E_2, G_2, F_2) → k_4
  108. Mirror(A, i_4) → A'_3
  109. Mirror(B, i_4) → B'_2
  110. Mirror(C_2, i_4) → H_2
  111. Mirror(D_2, i_4) → I_2
  112. Polygon(A'_3, B'_2, H_2, I_2) → mnogokut10', a'_3, b'_2, h_5, i_5
  113. Point(i_4) → J_2
  114. Point(i_4) → K_2
  115. Circle(J, Segment[J', L_2]) → p_4
  116. Circle(L_2, Segment[J, J']) → q_4
  117. Intersect(p_4, q_4, 1) → M_2
  118. Polygon(J, J', L_2, M_2) → mnogokut10_1, j_5, j'_2, l_4, m_4
  119. Segment(J, L_2) → n_4
  120. Segment(J', M_2) → r_4
  121. Intersect(n_4, r_4) → N_2
  122. Line(N_2, i_4) → s_4
  123. Point(s_4) → O_2
  124. Point(s_4) → P_2
  125. Segment(K_2, J_2) → f_5
  126. Segment(P_2, O_2) → g_5
  127. Polygon(A, B, R_2, Q_2) → mnogokut11, a_5, b_5, r_5, q_5
  128. Mirror(A, l) → A'_4
  129. Mirror(B, l) → B'_3
  130. Mirror(R_2, l) → S_2
  131. Mirror(Q_2, l) → U_2
  132. Polygon(A'_4, B'_3, S_2, U_2) → mnogokut11', a'_4, b'_3, s_5, u_2
  133. Polygon(J, J', W_2, V_2) → mnogokut12, j_6, j'_3, w_2, v_2
  134. Circle(A, 3) → e_5
  135. Intersect(e_5, r_2, 2) → Z_2
  136. Circle(Z_2, Segment[A, B]) → k_5
  137. Circle(B, Segment[A, Z_2]) → p_5
  138. Intersect(k_5, p_5, 1) → A_3
  139. Polygon(A, B, A_3, Z_2) → mnogokut13, a_6, b_6, a_7, z_2
  140. Segment(A, A_3) → l_5
  141. Segment(B, Z_2) → m_5
  142. Intersect(l_5, m_5) → B_3
  143. Circle(B_3, A_3) → t_5
  144. Point(t_5) → C_3
  145. Line(C_3, B_3) → n_5
  146. Intersect(t_5, n_5, 2) → D_3
  147. Line(J, r_2) → f_6
  148. CircleSector(B_3, D_3, C_3) → c_6
  149. Mirror(A, n_5) → A'_5
  150. Mirror(B, n_5) → B'_4
  151. Mirror(A_3, n_5) → E_3
  152. Mirror(Z_2, n_5) → F_3
  153. Polygon(A'_5, B'_4, E_3, F_3) → mnogokut13', a'_5, b'_4, e_6, f_7
  154. Point(n_5) → G_3
  155. Point(n_5) → H_3
  156. Circle(J, 3) → d_6
  157. Intersect(d_6, f_6, 2) → I_3
  158. Circle(I_3, Segment[J, J']) → g_6
  159. Circle(J', Segment[J, I_3]) → h_6
  160. Intersect(g_6, h_6, 1) → J_3
  161. Polygon(J, J', J_3, I_3) → mnogokut13_1, j_7, j'_4, j_8, i_6
  162. Segment(J, J_3) → k_6
  163. Segment(J', I_3) → l_6
  164. Intersect(k_6, l_6) → K_3
  165. Line(K_3, n_5) → m_6
  166. Point(m_6) → L_3
  167. Point(m_6) → M_3
  168. Segment(H_3, G_3) → n_6
  169. Segment(M_3, L_3) → p_6
  170. Line(E, yAxis) → q_6
  171. Intersect(c, q_6, 1) → N_3
  172. Rotate(N_3, 36°, E) → O_3
  173. Angle(N_3, E, O_3) → α
  174. Rotate(N_3, (-(36°)), E) → P_3
  175. Angle(P_3, E, N_3) → β
  176. Circle(O_3, Segment[P_3, O_3]) → r_6
  177. Intersect(c, r_6, 1) → Q_3
  178. Circle(Q_3, Segment[O_3, Q_3]) → s_6
  179. Intersect(c, s_6, 1) → R_3
  180. Circle(R_3, Segment[Q_3, R_3]) → t_6
  181. Intersect(c, t_6, 1) → S_3
  182. Polygon(P_3, O_3, Q_3, R_3, S_3) → mnogokut14, p_7, o_3, q_7, r_7, s_7
  183. Mirror(P_3, l) → U_3
  184. Mirror(O_3, l) → V_3
  185. Mirror(Q_3, l) → W_3
  186. Mirror(R_3, l) → Z_3
  187. Mirror(S_3, l) → A_4
  188. Polygon(U_3, V_3, W_3, Z_3, A_4) → mnogokut14', u_3, v_3, w_3, z_3, a_8
  189. Translate(P_3, Vector[A, J]) → B_4
  190. Translate(O_3, Vector[A, J]) → C_4
  191. Translate(Q_3, Vector[A, J]) → D_4
  192. Translate(R_3, Vector[A, J]) → E_4
  193. Translate(S_3, Vector[A, J]) → F_4
  194. Polygon(B_4, C_4, D_4, E_4, F_4) → mnogokut14'_1, b_7, c_7, d_7, e_7, f_8
  195. Rotate(N_3, 30°, E) → G_4
  196. Angle(N_3, E, G_4) → γ
  197. Rotate(N_3, (-(30°)), E) → H_4
  198. Angle(H_4, E, N_3) → δ
  199. Circle(G_4, Segment[H_4, G_4]) → g_7
  200. Intersect(c, g_7, 1) → I_4
  201. Circle(I_4, Segment[G_4, I_4]) → h_7
  202. Intersect(c, h_7, 1) → J_4
  203. Circle(J_4, Segment[I_4, J_4]) → k_7
  204. Intersect(c, k_7, 1) → K_4
  205. Circle(K_4, Segment[J_4, K_4]) → t_7
  206. Intersect(c, t_7, 1) → L_4
  207. Polygon(H_4, G_4, I_4, J_4, K_4, L_4) → mnogokut15, h_8, g_8, i_7, j_9, k_8, l_7
  208. Mirror(H_4, l) → M_4
  209. Mirror(G_4, l) → N_4
  210. Mirror(I_4, l) → O_4
  211. Mirror(J_4, l) → P_4
  212. Mirror(K_4, l) → Q_4
  213. Mirror(L_4, l) → R_4
  214. Polygon(M_4, N_4, O_4, P_4, Q_4, R_4) → mnogokut15', m_7, n_7, o_4, p_8, q_8, r_8
  215. Translate(H_4, Vector[A, J]) → S_4
  216. Translate(G_4, Vector[A, J]) → U_4
  217. Translate(I_4, Vector[A, J]) → V_4
  218. Translate(J_4, Vector[A, J]) → W_4
  219. Translate(K_4, Vector[A, J]) → Z_4
  220. Translate(L_4, Vector[A, J]) → A_5
  221. Polygon(S_4, U_4, V_4, W_4, Z_4, A_5) → mnogokut15'_1, s_8, u_4, v_4, w_4, z_4, a_9
  222. Polygon(B_5, C_5, D_5, L_5, M_5, E_5, I_5, K_5, H_5, F_5, J_5, G_5) → mnogokut16, b_8, g_9, d_8, l_8, c_8, m_8, e_8, i_8, k_9, h_9, f_9, j_{10}
  223. Mirror(B_5, l) → N_5
  224. Mirror(C_5, l) → O_5
  225. Mirror(M_5, l) → P_5
  226. Mirror(E_5, l) → Q_5
  227. Mirror(I_5, l) → R_5
  228. Mirror(K_5, l) → S_5
  229. Mirror(H_5, l) → T_5
  230. Mirror(F_5, l) → U_5
  231. Mirror(J_5, l) → V_5
  232. Mirror(G_5, l) → W_5
  233. Mirror(D_5, l) → Z_5
  234. Mirror(L_5, l) → A_6
  235. Polygon(N_5, O_5, Z_5, A_6, P_5, Q_5, R_5, S_5, T_5, U_5, V_5, W_5) → mnogokut16', n_8, w_5, z_5, a_{10}, o_5, p_9, q_9, r_9, s_9, t_8, u_5, v_5
  236. Translate(B_5, Vector[A, J]) → B_6
  237. Translate(C_5, Vector[A, J]) → C_6
  238. Translate(M_5, Vector[A, J]) → D_6
  239. Translate(E_5, Vector[A, J]) → E_6
  240. Translate(I_5, Vector[A, J]) → F_6
  241. Translate(K_5, Vector[A, J]) → G_6
  242. Translate(H_5, Vector[A, J]) → H_6
  243. Translate(F_5, Vector[A, J]) → I_6
  244. Translate(J_5, Vector[A, J]) → J_6
  245. Translate(G_5, Vector[A, J]) → K_6
  246. Translate(D_5, Vector[A, J]) → L_6
  247. Translate(L_5, Vector[A, J]) → M_6
  248. Polygon(B_6, C_6, L_6, M_6, D_6, E_6, F_6, G_6, H_6, I_6, J_6, K_6) → mnogokut16'_1, b_9, k_{10}, l_9, m_9, c_9, d_9, e_9, f_{10}, g_{10}, h_{10}, i_9, j_{11}
  249. Polygon(A, P_6, N_6, D, S_6, U_6, Q_6, B, C, O_6, T_6, R_6) → mnogokut17, a_{11}, r_{10}, n_9, d_{10}, p_{10}, s_{10}, u_6, q_{10}, b_{10}, c_{10}, o_6, t_9
  250. Mirror(A, l) → A'_6
  251. Mirror(P_6, l) → V_6
  252. Mirror(S_6, l) → W_6
  253. Mirror(U_6, l) → Z_6
  254. Mirror(Q_6, l) → A_7
  255. Mirror(B, l) → B'_5
  256. Mirror(C, l) → C'
  257. Mirror(O_6, l) → B_7
  258. Mirror(T_6, l) → C_7
  259. Mirror(R_6, l) → D_7
  260. Mirror(N_6, l) → E_7
  261. Mirror(D, l) → D'
  262. Polygon(A'_6, V_6, E_7, D', W_6, Z_6, A_7, B'_5, C', B_7, C_7, D_7) → mnogokut17', a'_6, d_{11}, e_{10}, d', v_6, w_6, z_6, a_{12}, b'_5, c', b_{11}, c_{11}
  263. Translate(A, Vector[A, J]) → A'_7
  264. Translate(P_6, Vector[A, J]) → F_7
  265. Translate(S_6, Vector[A, J]) → G_7
  266. Translate(U_6, Vector[A, J]) → H_7
  267. Translate(Q_6, Vector[A, J]) → I_7
  268. Translate(B, Vector[A, J]) → B'_6
  269. Translate(C, Vector[A, J]) → C'_1
  270. Translate(O_6, Vector[A, J]) → J_7
  271. Translate(T_6, Vector[A, J]) → K_7
  272. Translate(R_6, Vector[A, J]) → L_7
  273. Translate(N_6, Vector[A, J]) → M_7
  274. Translate(D, Vector[A, J]) → D'_1
  275. Polygon(A'_7, F_7, M_7, D'_1, G_7, H_7, I_7, B'_6, C'_1, J_7, K_7, L_7) → mnogokut17'_1, a'_7, l_{10}, m_{10}, d'_1, f_{11}, g_{11}, h_{11}, i_{10}, b'_6, c'_1, j_{12}, k_{11}
  276. Polygon(R_7, S_7, Q_7, O_7, C, D, N_7, P_7) → mnogokut18, r_{11}, s_{11}, q_{11}, o_7, c_{12}, d_{12}, n_{10}, p_{11}
  277. Mirror(R_7, l) → T_7
  278. Mirror(S_7, l) → U_7
  279. Mirror(Q_7, l) → V_7
  280. Mirror(O_7, l) → W_7
  281. Mirror(C, l) → C'_2
  282. Mirror(D, l) → D'_2
  283. Mirror(N_7, l) → Z_7
  284. Mirror(P_7, l) → A_8
  285. Polygon(T_7, U_7, V_7, W_7, C'_2, D'_2, Z_7, A_8) → mnogokut18', t_{10}, u_7, v_7, w_7, c'_2, d'_2, z_7, a_{13}
  286. Translate(R_7, Vector[A, J]) → B_8
  287. Translate(S_7, Vector[A, J]) → C_8
  288. Translate(Q_7, Vector[A, J]) → D_8
  289. Translate(O_7, Vector[A, J]) → E_8
  290. Translate(C, Vector[A, J]) → C'_3
  291. Translate(D, Vector[A, J]) → D'_3
  292. Translate(N_7, Vector[A, J]) → F_8
  293. Translate(P_7, Vector[A, J]) → G_8
  294. Polygon(B_8, C_8, D_8, E_8, C'_3, D'_3, F_8, G_8) → mnogokut18'_1, b_{12}, c_{13}, d_{13}, e_{11}, c'_3, d'_3, f_{12}, g_{12}
  295. Polygon(D, H_8, A, O_8, M_8, I_8, C, N_8, B, K_8, L_8, J_8) → mnogokut19, d_{14}, j_{13}, a_{14}, o_8, h_{12}, m_{11}, i_{11}, c_{14}, n_{11}, b_{13}, k_{12}, l_{11}
  296. Mirror(D, l) → D'_4
  297. Mirror(H_8, l) → P_8
  298. Mirror(M_8, l) → Q_8
  299. Mirror(I_8, l) → R_8
  300. Mirror(C, l) → C'_4
  301. Mirror(N_8, l) → S_8
  302. Mirror(B, l) → B'_7
  303. Mirror(K_8, l) → T_8
  304. Mirror(L_8, l) → U_8
  305. Mirror(J_8, l) → V_8
  306. Mirror(A, l) → A'_8
  307. Mirror(O_8, l) → W_8
  308. Polygon(D'_4, P_8, A'_8, W_8, Q_8, R_8, C'_4, S_8, B'_7, T_8, U_8, V_8) → mnogokut19', d'_4, v_8, a'_8, w_8, p_{12}, q_{12}, r_{12}, c'_4, s_{12}, b'_7, t_{11}, u_8
  309. Translate(D, Vector[A, J]) → D'_5
  310. Translate(H_8, Vector[A, J]) → Z_8
  311. Translate(M_8, Vector[A, J]) → A_9
  312. Translate(I_8, Vector[A, J]) → B_9
  313. Translate(C, Vector[A, J]) → C'_5
  314. Translate(N_8, Vector[A, J]) → C_9
  315. Translate(B, Vector[A, J]) → B'_8
  316. Translate(K_8, Vector[A, J]) → D_9
  317. Translate(L_8, Vector[A, J]) → E_9
  318. Translate(J_8, Vector[A, J]) → F_9
  319. Translate(A, Vector[A, J]) → A'_9
  320. Translate(O_8, Vector[A, J]) → G_9
  321. Polygon(D'_5, Z_8, A'_9, G_9, A_9, B_9, C'_5, C_9, B'_8, D_9, E_9, F_9) → mnogokut19'_1, d'_5, f_{13}, a'_9, g_{13}, z_8, a_{15}, b_{14}, c'_5, c_{15}, b'_8, d_{15}, e_{12}
  322. Textfield(brOsiUpisano) → txtPolje1
  323. Element(brojOsiSimetrije, 14) → brOsi

AI 绘图提示语参考

绘制一个由正方形和圆弧组合而成的字母X形状。先画一个正方形，连接其对角线，交点作为圆心，以正方形顶点为半径画圆，在圆上取一点，过该点和圆心作直线，与圆交于另一点，用这两点与圆心构成一个半圆。再画一个较小的正方形，连接其对角线，交点与之前半圆的圆心连线作为对称轴。在对称轴上取两点作线段，并画一个三角形，使其关于对称轴镜像对称。最后画一个等边三角形，作其两条边的垂直平分线，交点作为圆心，以三角形顶点为半径画圆，在圆上取点作直线，形成半圆，并让该三角形关于此直线镜像。整体图形包含多个对称轴，需清晰展示各部分的镜像关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。