四边形的对称轴

四边形的对称轴
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2020-07-29 23:30
①本范例演示如何寻找四边形的对称轴;②涉及的核心数学概念包括对称性、线段、角度、距离、反射变换;③作图要点——从下方『构造摘要』提炼用到了 Line, Segment, Angle, Distance, Point, Mirror, Intersect, Polygon, Circle, OrthogonalLine 等GeoGebra命令/对象;④适合初中及以上学生学习和探索几何对称性。

GeoGebra 命令及参数

  1. Line(D, C) → f
  2. Line(B, C) → g
  3. Segment(B, C) → j
  4. Segment(C, D) → k
  5. Line(B, D) → m
  6. Segment(E, F) → p
  7. Segment(B, D) → t
  8. Angle(B, C, D) → γ
  9. Distance(B, C) → distanceBC
  10. Distance(C, D) → distanceCD
  11. Distance(D, B) → distanceDB
  12. Line(J, K) → b_1
  13. Point(b_1) → L
  14. Point(b_1) → N
  15. Point(b_1) → O
  16. Point(b_1) → Q
  17. Mirror(M, b_1) → M'
  18. Point(b_1) → P
  19. Point(b_1) → R
  20. Segment(P, M) → c_1
  21. Segment(M, R) → f_1
  22. Segment(R, M') → g_1
  23. Segment(M', P) → h_1
  24. Segment(P, R) → i_1
  25. Segment(M, M') → j_1
  26. Intersect(i_1, j_1) → S
  27. Line(P, M') → k_1
  28. Line(M, R) → l_1
  29. Line(R, M') → m_1
  30. Line(M, P) → n_1
  31. Polygon(P, M, R, M') → poly3, p_1, m_2, r_1, m'
  32. Distance(P, M) → distancePM
  33. Distance(M, R) → distanceMR
  34. Distance(R, M') → distanceRM'
  35. Distance(M', P) → distanceM'P
  36. Angle(M', P, M) → ζ
  37. Angle(P, M, R) → η
  38. Angle(M, R, M') → θ
  39. Angle(R, M', P) → ι
  40. Angle(P, S, M') → κ
  41. Distance(M', M) → distanceM'M
  42. Distance(P, R) → distancePR
  43. Segment(T, U) → d_1
  44. Circle(S, Segment[U, T]) → e_1
  45. Point(e_1) → V
  46. Line(V, S) → q_1
  47. Mirror(P, q_1) → P'
  48. Mirror(M, q_1) → M'_1
  49. Mirror(R, q_1) → R'
  50. Mirror(M', q_1) → M''
  51. Polygon(P', M'_1, R', M'') → poly3', p', m'_1, r', m''
  52. Line(W, Z) → s_1
  53. Point(s_1) → B_1
  54. Point(s_1) → C_1
  55. Segment(B_1, C_1) → a_2
  56. Point(s_1) → D_1
  57. Point(s_1) → E_1
  58. Segment(D_1, E_1) → b_2
  59. Point(a_2) → H_1
  60. Line(A_1, F_1) → t_1
  61. Line(G_1, t_1) → c_2
  62. Point(c_2) → I_1
  63. Line(A_1, G_1) → h_2
  64. Line(F_1, I_1) → i_2
  65. Line(A_1, F_1) → j_2
  66. Line(I_1, G_1) → k_2
  67. Distance(G_1, A_1) → distanceG1A1
  68. Distance(A_1, F_1) → distanceA1F1
  69. Distance(F_1, I_1) → distanceF1I1
  70. Distance(I_1, G_1) → distanceI1G1
  71. Angle(I_1, G_1, A_1) → λ
  72. Angle(G_1, A_1, F_1) → μ
  73. Angle(A_1, F_1, I_1) → ν
  74. Angle(F_1, I_1, G_1) → ξ
  75. Segment(A_1, I_1) → l_2
  76. Segment(G_1, F_1) → n_2
  77. Intersect(l_2, n_2) → K_1
  78. Distance(G_1, F_1) → distanceG1F1
  79. Distance(A_1, I_1) → distanceA1I1
  80. Angle(G_1, K_1, I_1) → ο
  81. Polygon(G_1, A_1, F_1, I_1) → poly4, g_2, a_3, f_2, i_3
  82. Segment(J_1, L_1) → d_2
  83. Circle(K_1, Segment[J_1, L_1]) → e_2
  84. Point(e_2) → M_1
  85. Line(K_1, M_1) → p_2
  86. Mirror(G_1, p_2) → N_1
  87. Mirror(A_1, p_2) → O_1
  88. Mirror(F_1, p_2) → P_1
  89. Mirror(I_1, p_2) → Q_1
  90. Polygon(N_1, O_1, P_1, Q_1) → poly4', n_3, o_1, p_3, q_2
  91. Line(R_1, S_1) → r_2
  92. Line(T_1, r_2) → s_2
  93. Segment(R_1, T_1) → t_2
  94. Circle(S_1, Segment[R_1, S_1]) → c_3
  95. Circle(S_1, Segment[R_1, T_1]) → d_3
  96. Circle(R_1, Segment[T_1, S_1]) → e_3
  97. Intersect(d_3, e_3, 2) → U_1
  98. Segment(T_1, U_1) → b_3
  99. Segment(U_1, S_1) → f_3
  100. Segment(S_1, R_1) → g_3
  101. Intersect(t_2, f_3) → V_1
  102. Intersect(b_3, g_3) → W_1
  103. Distance(U_1, V_1) → distanceU1V1
  104. Distance(U_1, W_1) → distanceU1W1
  105. Intersect(f_2, g_2) → Z_1
  106. Polygon(R_1, T_1, U_1, S_1) → poly5, r_3, t_3, u_1, s_3
  107. Line(R_1, T_1) → h_3
  108. Line(S_1, U_1) → j_3
  109. Line(T_1, U_1) → k_3
  110. Line(R_1, S_1) → l_3
  111. Segment(T_1, S_1) → m_3
  112. Segment(R_1, U_1) → q_3
  113. Intersect(m_3, q_3) → A_2
  114. Segment(B_2, C_2) → a_4
  115. Circle(A_2, Segment[B_2, C_2]) → c_4
  116. Point(c_4) → D_2
  117. Line(A_2, D_2) → b_4
  118. Mirror(R_1, b_4) → E_2
  119. Mirror(T_1, b_4) → F_2
  120. Mirror(U_1, b_4) → G_2
  121. Mirror(S_1, b_4) → H_2
  122. Polygon(E_2, F_2, G_2, H_2) → poly5', e_4, f_4, g_4, h_4
  123. Distance(R_1, T_1) → distanceR1T1
  124. Distance(T_1, U_1) → distanceT1U1
  125. Distance(U_1, S_1) → distanceU1S1
  126. Distance(S_1, R_1) → distanceS1R1
  127. Distance(R_1, U_1) → distanceR1U1
  128. Distance(T_1, S_1) → distanceT1S1
  129. Angle(R_1, A_2, S_1) → ρ
  130. Angle(S_1, R_1, T_1) → σ
  131. Angle(R_1, T_1, U_1) → τ
  132. Angle(T_1, U_1, S_1) → υ
  133. Angle(U_1, S_1, R_1) → ϕ
  134. Line(I_2, J_2) → d_4
  135. Line(K_2, d_4) → i_4
  136. Line(K_2, I_2) → j_4
  137. Line(J_2, j_4) → k_4
  138. Intersect(i_4, k_4) → L_2
  139. Polygon(K_2, I_2, J_2, L_2) → poly6, k_5, i_5, j_5, l_4
  140. Line(I_2, K_2) → m_4
  141. Line(J_2, L_2) → n_4
  142. Line(I_2, J_2) → p_4
  143. Line(K_2, L_2) → q_4
  144. Segment(K_2, J_2) → r_4
  145. Segment(I_2, L_2) → s_4
  146. Intersect(r_4, s_4) → M_2
  147. Segment(N_2, O_2) → t_4
  148. Circle(M_2, Segment[N_2, O_2]) → c_5
  149. Point(c_5) → P_2
  150. Line(M_2, P_2) → a_5
  151. Mirror(K_2, a_5) → Q_2
  152. Mirror(I_2, a_5) → R_2
  153. Mirror(J_2, a_5) → S_2
  154. Mirror(L_2, a_5) → T_2
  155. Polygon(Q_2, R_2, S_2, T_2) → poly6', q_5, r_5, s_5, t_5
  156. Distance(I_2, K_2) → distanceI2K2
  157. Distance(K_2, L_2) → distanceK2L2
  158. Distance(L_2, J_2) → distanceL2J2
  159. Distance(J_2, I_2) → distanceJ2I2
  160. Distance(I_2, L_2) → distanceI2L2
  161. Distance(K_2, J_2) → distanceK2J2
  162. Angle(J_2, I_2, K_2) → χ
  163. Angle(I_2, K_2, L_2) → ψ
  164. Angle(K_2, L_2, J_2) → ω
  165. Angle(L_2, J_2, I_2) → α_1
  166. Angle(I_2, M_2, J_2) → β_1
  167. Distance(I_2, J_2) → distanceI2J2
  168. Distance(I_2, L_2) → v_1
  169. Distance(K_2, J_2) → w_1
  170. Circle(U_2, V_2) → d_5
  171. Point(d_5) → W_2
  172. Segment(V_2, W_2) → b_5
  173. Mirror(U_2, b_5) → Z_2
  174. Polygon(W_2, U_2, V_2, Z_2) → poly7, w_2, u_2, v_2, z_2
  175. Segment(U_2, Z_2) → e_5
  176. Intersect(b_5, e_5) → A_3
  177. Distance(W_2, V_2) → distanceW2V2
  178. Distance(U_2, Z_2) → distanceU2Z2
  179. Angle(W_2, A_3, Z_2) → γ_1
  180. Angle(Z_2, W_2, U_2) → δ_1
  181. Angle(W_2, U_2, V_2) → ε_1
  182. Angle(U_2, V_2, Z_2) → ζ_1
  183. Angle(V_2, Z_2, W_2) → η_1
  184. Distance(W_2, U_2) → distanceW2U2
  185. Distance(U_2, V_2) → distanceU2V2
  186. Distance(V_2, Z_2) → distanceV2Z2
  187. Distance(Z_2, W_2) → distanceZ2W2
  188. Segment(B_3, C_3) → f_5
  189. Circle(A_3, Segment[B_3, C_3]) → g_5
  190. Point(g_5) → D_3
  191. Line(D_3, A_3) → h_5
  192. Mirror(W_2, h_5) → E_3
  193. Mirror(U_2, h_5) → F_3
  194. Mirror(V_2, h_5) → G_3
  195. Mirror(Z_2, h_5) → H_3
  196. Polygon(E_3, F_3, G_3, H_3) → poly7', e_6, f_6, g_6, h_6
  197. Line(W_2, U_2) → l_5
  198. Line(U_2, V_2) → m_5
  199. Line(V_2, Z_2) → n_5
  200. Line(Z_2, W_2) → p_5
  201. Segment(I_3, J_3) → a_6
  202. OrthogonalLine(I_3, a_6) → b_6
  203. OrthogonalLine(J_3, a_6) → c_6
  204. Point(b_6) → K_3
  205. Line(K_3, b_6) → d_6
  206. Line(K_3, d_6) → i_6
  207. Line(K_3, a_6) → j_6
  208. Intersect(c_6, j_6) → L_3
  209. Polygon(I_3, K_3, L_3, J_3) → poly8, i_7, k_6, l_6, j_7
  210. Distance(I_3, K_3) → distanceI3K3
  211. Distance(K_3, L_3) → distanceK3L3
  212. Distance(L_3, J_3) → distanceL3J3
  213. Distance(J_3, I_3) → distanceJ3I3
  214. Angle(J_3, I_3, K_3) → θ_1
  215. Angle(I_3, K_3, L_3) → ι_1
  216. Angle(K_3, L_3, J_3) → κ_1
  217. Angle(L_3, J_3, I_3) → λ_1
  218. Segment(K_3, J_3) → m_6
  219. Segment(I_3, L_3) → n_6
  220. Intersect(m_6, n_6) → M_3
  221. Line(I_3, K_3) → p_6
  222. Line(K_3, L_3) → q_6
  223. Line(L_3, J_3) → r_6
  224. Line(J_3, I_3) → s_6
  225. Angle(I_3, M_3, J_3) → μ_1
  226. Distance(I_3, L_3) → distanceI3L3
  227. Distance(K_3, J_3) → distanceK3J3
  228. Segment(N_3, O_3) → t_6
  229. Circle(M_3, Segment[N_3, O_3]) → c_7
  230. Point(c_7) → P_3
  231. Line(M_3, P_3) → a_7
  232. Mirror(I_3, a_7) → Q_3
  233. Mirror(K_3, a_7) → R_3
  234. Mirror(L_3, a_7) → S_3
  235. Mirror(J_3, a_7) → T_3
  236. Polygon(Q_3, R_3, S_3, T_3) → poly8', q_7, r_7, s_7, t_7
  237. Segment(A, G) → a
  238. OrthogonalLine(A, a) → b
  239. OrthogonalLine(G, a) → c
  240. Circle(A, Segment[A, G]) → d
  241. Circle(G, Segment[A, G]) → e
  242. Intersect(d, b, 2) → H
  243. Intersect(e, c, 2) → I
  244. Polygon(A, H, I, G) → poly1, a_1, h, i, g_7
  245. Segment(H, G) → l
  246. Segment(I, A) → n
  247. Intersect(l, n) → U_3
  248. Line(H, I) → q
  249. Line(G, A) → r
  250. Line(A, H) → b_7
  251. Line(I, G) → d_7
  252. Angle(G, U_3, I) → α
  253. Angle(G, A, H) → β
  254. Angle(A, H, I) → δ
  255. Angle(H, I, G) → ε
  256. Angle(I, G, A) → ν_1
  257. Distance(A, H) → distanceAH
  258. Distance(H, I) → distanceHI
  259. Distance(I, G) → distanceIG
  260. Distance(G, A) → distanceGA
  261. Distance(A, I) → distanceAI
  262. Distance(H, G) → distanceHG
  263. Segment(B, V_3) → e_7
  264. Segment(V_3, D) → f_7
  265. Polygon(V_3, B, C, D) → poly2, v_3, b_8, c_8, d_8
  266. Line(B, V_3) → h_7
  267. Line(V_3, D) → k_7
  268. Segment(C, V_3) → l_7
  269. Intersect(t, l_7) → W_3
  270. Intersect(c_8, v_3) → Z_3
  271. Intersect(d_8, b_8) → A_4
  272. Segment(B_4, C_4) → m_7
  273. Circle(W_3, Segment[B_4, C_4]) → p_7
  274. Point(p_7) → D_4
  275. Line(W_3, D_4) → n_7
  276. Mirror(V_3, n_7) → E_4
  277. Mirror(B, n_7) → B'
  278. Mirror(C, n_7) → C'
  279. Mirror(D, n_7) → D'
  280. Polygon(E_4, B', C', D') → poly2', e_8, b', c', d'
  281. Mirror(B', b') → B''
  282. Mirror(C', b') → C''
  283. Distance(V_3, B) → distanceV3B
  284. Distance(B, C) → z_1
  285. Distance(C, D) → o_2
  286. Distance(D, V_3) → distanceDV3
  287. Distance(V_3, C) → distanceV3C
  288. Distance(B, D) → distanceBD
  289. Angle(D, V_3, B) → ξ_1
  290. Angle(V_3, B, C) → ο_1
  291. Angle(B, C, D) → ρ_1
  292. Angle(C, D, V_3) → σ_1
  293. Angle(V_3, W_3, D) → τ_1
  294. Segment(F_4, G_4) → a_8
  295. Circle(U_3, Segment[F_4, G_4]) → f_8
  296. Point(f_8) → H_4
  297. Line(U_3, H_4) → g_8
  298. Mirror(A, g_8) → A'
  299. Mirror(H, g_8) → H'
  300. Mirror(I, g_8) → I'
  301. Mirror(G, g_8) → G'
  302. Polygon(A', H', I', G') → poly1', a', h', i', g'

AI 绘图提示语参考

画一个四边形,并找出它的对称轴。首先画出四边形的四条边和两条对角线,然后测量四边形的边长和角度。接着,找到对角线的中点,并画出其中一条对角线的垂线,这条垂线就是四边形的一条对称轴。最后,通过镜像变换,验证对称轴的对称性。
以上提示语为根据原始绘图命令整理而来,仅供参考。