Phase F: Octagon

Phase F: Octagon
几何画板及绘图内容加载中,请稍等……
2023-01-09 22:42
这个范例展示了正八边形的构造与性质。涉及的核心数学概念包括多边形、角度、变换和优化问题。作图要点包括使用Ray、Circle、Intersect等GeoGebra命令,以及Polygon、Point等对象。适合初中及以上学生用于几何学习与作图实践。

GeoGebra 命令及参数

  1. Ray(S_1, D) → i_5
  2. Circle(D, 3) → k
  3. Intersect(k, i_5) → U_1, V_1
  4. Ray(D, A_1) → a_3
  5. Intersect(k, a_3, 1) → D_1
  6. Segment(C, A_1) → i_3
  7. Circle(C, 3) → h
  8. Intersect(h, i_3) → L,
  9. Ray(C, D) → c
  10. Intersect(h, c, 1) → P
  11. Ray(A, B) → a
  12. Ray(B, C) → b
  13. Ray(E, A) → e
  14. Circle(A, 3) → f
  15. Circle(B, 3) → g
  16. Circle(E, 3) → p
  17. Segment(C, A) → i
  18. Segment(C, E) → j
  19. Intersect(f, a, 1) → F
  20. Intersect(f, i, 1) → G
  21. Intersect(f, e, 1) → H
  22. Intersect(f, e, 2) → I
  23. Intersect(p, e) → J,
  24. Intersect(p, j) → K,
  25. Intersect(k, c, 2) → O
  26. Intersect(h, j, 1) → Q
  27. Intersect(h, i, 1) → R
  28. Intersect(h, b, 1) → S
  29. Intersect(g, a, 2) → T
  30. Intersect(g, b, 1) → U
  31. Intersect(g, a, 1) → V
  32. CircleSector(A, F, G) → q
  33. CircleSector(A, G, H) → r
  34. CircleSector(B, U, V) → s
  35. CircleSector(C, Q, R) → c_1
  36. CircleSector(C, R, S) → d_1
  37. Intersect(k, c, 1) → W
  38. CircleSector(E, J, K) → h_1
  39. CircleSector(A, I, F) → p_1
  40. CircleSector(B, T, U) → q_1
  41. Intersect(h, b, 2) → Z
  42. Vector(A, B) → u
  43. Vector(C, B) → v
  44. Vector(E, D) → w
  45. Vector(C, D) → u_1
  46. Vector(C, E) → v_1
  47. Vector(A, E) → w_1
  48. Translate(d_1, v) → s_1
  49. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  50. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], B) → C'
  51. Dilate(A, If[1 < l ≤ 2, 2 - l, 0], E) → A'_1
  52. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  53. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  54. Dilate(E, If[2 < l ≤ 3, 3 - l, 0], D) → E'
  55. Vector(A, B) → a_1
  56. Vector(C, C') → b_1
  57. Vector(A, A'_1) → i_1
  58. Vector(E, E') → j_1
  59. Vector(C, C'_2) → l_1
  60. Vector(C, C'_1) → m_1
  61. Translate(d_1, b_1) → c_2
  62. Translate(r, i_1) → r'_1
  63. Translate(r, w_1) → r'_2
  64. Translate(c_1, v_1) → f_2
  65. Translate(c_1, m_1) → g_2
  66. CircleSector(C, P, L) → t
  67. Translate(t, u_1) → t'
  68. Translate(t, l_1) → t'_1
  69. Translate(q, a_1) → r'
  70. Translate(d_1, b_1) → h_2
  71. Vector(A, A') → n_1
  72. Mirror(r'_2, E) → p_2
  73. Vector(D, C) → u_2
  74. Vector(E, C) → v_2
  75. Vector(A, C) → w_2
  76. Vector(B, C) → l_2
  77. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  78. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  79. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  80. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  81. Vector(B, B') → m_2
  82. Vector(A, A'_2) → n_2
  83. Vector(E, E'_1) → s_2
  84. Vector(D, D') → t_2
  85. Translate(q_1, m_2) → c_3
  86. Translate(q_1, l_2) → d_3
  87. Translate(p_1, w_2) → e_3
  88. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  89. Ray(A_1, E) → b_3
  90. Circle(A_1, 3) → q_3
  91. Intersect(q_3, i_3) → M,
  92. Intersect(q_3, b_3, 1) → N
  93. Intersect(q_3, a_3, 2) → B_1
  94. Intersect(k, c, 2) → C_1
  95. Intersect(k, c, 1) → E_1
  96. CircleSector(C, L, Q) → d
  97. Intersect(p, b_3, 1) → F_1
  98. Dilate(E, If[2 < l < 3, 3 - l, 0], A_1) → E'_2
  99. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  100. Dilate(C, If[3 < l < 4, 4 - l, 0], D) → C'_4
  101. Dilate(A_1, If[3 < l < 4, 4 - l, 0], D) → H_1
  102. Vector(E, A_1) → f_1
  103. Vector(C, A_1) → g_1
  104. Translate(d, g_1) → d'
  105. Vector(E, E'_2) → a_2
  106. Vector(C, C'_3) → b_2
  107. Translate(d, b_2) → d'_1
  108. CircleSector(A_1, N, M) → f_3
  109. Intersect(q_3, a_3, 1) → I_1
  110. CircleSector(A_1, M, I_1) → g_3
  111. CircleSector(D, D_1, E_1) → k_3
  112. Vector(C, D) → j_2
  113. Vector(A_1, D) → u_3
  114. Translate(g_3, u_3) → p_3
  115. Translate(t, j_2) → t'_2
  116. Mirror(p_3, D) → r_3
  117. Vector(A_1, H_1) → v_3
  118. Vector(C, C'_4) → w_3
  119. Translate(g_3, v_3) → s_3
  120. Translate(t, w_3) → t'_3
  121. Intersect(p, b_3, 2) → J_1
  122. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  123. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  124. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  125. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  126. Intersect(p, e, 1) → L_1
  127. CircleSector(E, J_1, L_1) → f_4
  128. CircleSector(A_1, B_1, N) → g_4
  129. Vector(E, C) → j_3
  130. Vector(A_1, C) → l_3
  131. Vector(D, C) → m_3
  132. Translate(f_4, j_3) → k_4
  133. Translate(g_4, l_3) → p_4
  134. CircleSector(D, V_1, D_1) → h_4
  135. Translate(h_4, m_3) → q_4
  136. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  137. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_1
  138. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  139. Vector(E, E'_3) → n_3
  140. Vector(A_1, M_1) → u_4
  141. Vector(D, D'_1) → v_4
  142. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  143. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  144. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  145. Ray(C, N_1) → i_4
  146. Ray(N_1, D) → l_4
  147. Circle(N_1, 3) → d_5
  148. Intersect(d_5, i_4, 2) → K_1
  149. Intersect(d_5, l_4, 1) → O_1
  150. Intersect(d_5, i_4, 1) → P_1
  151. Intersect(h, i_4, 1) → Q_1
  152. Intersect(k, l_4, 1) → R_1
  153. Dilate(D, If[4 < l < 5, 5 - l, 0], N_1) → D'_2
  154. Dilate(C, If[4 < l < 5, 5 - l, 0], N_1) → C'_5
  155. Vector(C, N_1) → w_4
  156. Vector(D, N_1) → j_4
  157. Vector(D, D'_2) → m_4
  158. Vector(C, C'_5) → n_4
  159. Ray(N_1, S_1) → b_5
  160. Segment(D, C) → j_5
  161. Intersect(d_5, b_5) → T_1,
  162. Dilate(D, If[5 < l < 6, 6 - l, 0], S_1) → D'_3
  163. Dilate(N_1, If[5 < l < 6, 6 - l, 0], S_1) → W_1
  164. Vector(D, D'_3) → u_5
  165. Vector(N_1, W_1) → v_5
  166. Vector(D, S_1) → w_5
  167. Vector(N_1, S_1) → m_5
  168. Circle(S_1, 3) → r_6
  169. Intersect(r_6, i_5, 1) → Z_1
  170. Intersect(r_6, b_5) → A_2, B_2
  171. Polygon(A, B, C, N_1, S_1, D, A_1, E) → poly7, a_8, b_6, c_9, n_7, s_7, d_8, a_9, e_8
  172. CircleSector(C, Z, Q_1) → r_1
  173. CircleSector(W_1, B_2, Z_1) → h_7
  174. CircleSector(N_1, K_1, T_1) → k_7
  175. Vector(W_1, C) → u_6
  176. Vector(N_1, C) → v_6
  177. Translate(h_7, u_6) → p_7
  178. Vector(D, C) → w_6
  179. Translate(h_4, w_6) → q_7
  180. Dilate(W_1, If[5 < o < 6, 6 - o, 0], C) → C_2
  181. Vector(W_1, C_2) → i_6
  182. Translate(h_7, i_6) → r_7
  183. Dilate(N_1, If[6 < o < 7, 7 - o, 0], C) → D_2
  184. Vector(N_1, D_2) → j_6
  185. Translate(k_7, j_6) → t_7
  186. Vector(N_1, C) → l_6
  187. Translate(k_7, l_6) → f_8
  188. Ray(C, D) → m_6
  189. Ray(C, A_1) → b_7
  190. Ray(C, E) → i_7
  191. Point(i_7) → E_2
  192. Point(b_7) → F_2
  193. Point(m_6) → G_2
  194. Ray(E, E_2) → j_7
  195. Ray(A_1, F_2) → l_7
  196. Ray(D, G_2) → m_7
  197. Segment(C, S_1) → l_5
  198. Intersect(k, i_5, 1) → H_2
  199. Intersect(d_5, b_5, 1) → I_2
  200. CircleSector(N_1, I_2, P_1) → g_5
  201. Intersect(r_6, l_5) → J_2,
  202. Intersect(k, j_5) → K_2,
  203. CircleSector(S_1, Z_1, J_2) → f_7
  204. CircleSector(S_1, J_2, A_2) → f_5
  205. CircleSector(D, K_2, H_2) → h_5
  206. Intersect(h, j_5) → L_2,
  207. Intersect(h, l_5) → M_2,
  208. Intersect(h, i_4) → N_2,
  209. CircleSector(C, M_2, L_2) → k_5
  210. CircleSector(C, N_2, M_2) → p_5
  211. Dilate(D, If[4 < l < 5, 5 - l, 0], S_1) → D'_4
  212. Dilate(C, If[4 < l < 5, 5 - l, 0], S_1) → C'_6
  213. Vector(D, S_1) → n_5
  214. Vector(C, S_1) → q_5
  215. Translate(h_5, n_5) → r_5
  216. Vector(D, D'_4) → t_5
  217. Translate(h_5, t_5) → d_6
  218. Translate(k_5, q_5) → f_6
  219. Vector(C, C'_6) → g_6
  220. Translate(k_5, g_6) → h_6
  221. Dilate(S_1, If[5 < l < 6, 6 - l, 0], N_1) → O_2
  222. Dilate(C, If[5 < l < 6, 6 - l, 0], N_1) → C'_7
  223. Vector(C, N_1) → k_6
  224. Vector(C, C'_7) → n_6
  225. Vector(S_1, N_1) → s_6
  226. Translate(f_5, s_6) → t_6
  227. Vector(S_1, O_2) → u_7
  228. Translate(f_5, u_7) → e_7
  229. Ray(C, S_1) → b_8
  230. Polygon(D, C, S_1) → poly5, s_8, d_9, c_8
  231. Polygon(S_1, C, N_1) → poly6, n_8, s_9, c_{10}
  232. Segment(A, B) → g_8
  233. Segment(B, C) → h_8
  234. Segment(C, N_1) → i_8
  235. Segment(N_1, S_1) → j_8
  236. Segment(S_1, D) → k_8
  237. Segment(D, A_1) → l_8
  238. Segment(A_1, E) → m_8
  239. Segment(E, A) → p_8
  240. Midpoint(g_8) → P_2
  241. Midpoint(h_8) → Q_2
  242. Midpoint(i_8) → R_2
  243. Midpoint(j_8) → S_2
  244. Midpoint(p_8) → T_2
  245. Midpoint(m_8) → U_2
  246. Midpoint(l_8) → V_2
  247. Midpoint(k_8) → W_2
  248. Rotate(q, If[0 < l ≤ 1, (l * 180°), 180°], P_2) → q'_1
  249. Rotate(A, If[0 < l ≤ 1, (l * 180°), 180°], P_2) → A'_3
  250. Segment(P_2, A'_3) → k_2
  251. Rotate(d_1, If[0 < l ≤ 1, ((-l) * 180°), -180°], Q_2) → t_1
  252. Rotate(C, If[0 < l ≤ 1, ((-l) * 180°), -180°], Q_2) → C'_8
  253. Segment(Q_2, C'_8) → q_8
  254. Rotate(R, 3.141592653589793, Q_2) → R'
  255. Line(B, R') → r_8
  256. Rotate(r, If[1 < l ≤ 2, ((-(l - 1)) * 180°), -180°], T_2) → r'_3
  257. Rotate(A, If[1 < l ≤ 2, ((-(l - 1)) * 180°), -180°], T_2) → A'_4
  258. Segment(T_2, A'_4) → e_1
  259. CircleSector(E, K, F_1) → k_1
  260. Rotate(k_1, If[2 < l ≤ 3, ((-(l - 2)) * 180°), -180°], U_2) → d_2
  261. Rotate(E, If[2 < l ≤ 3, ((-(l - 2)) * 180°), -180°], U_2) → E'_4
  262. Segment(U_2, E'_4) → e_2
  263. Rotate(g_3, If[3 < l ≤ 4, ((-(l - 3)) * 180°), -180°], V_2) → q_2
  264. Rotate(A_1, If[3 < l ≤ 4, ((-(l - 3)) * 180°), -180°], V_2) → G_1
  265. Segment(V_2, G_1) → i_2
  266. Intersect(l_8, i_2) → Z_2
  267. Intersect(l_8, i_2) → A_3
  268. Rotate(h_5, If[4 < l ≤ 5, ((-(l - 4)) * 180°), -180°], W_2) → r_2
  269. Rotate(D, If[4 < l ≤ 5, ((-(l - 4)) * 180°), -180°], W_2) → D'_5
  270. Segment(W_2, D'_5) → t_3
  271. Translate(M_2, g_6) → B_3
  272. Ray(S_1, B_3) → s_5
  273. Rotate(p_5, If[5 < l ≤ 6, ((l - 5) * 180°), 180°], R_2) → e_6
  274. Rotate(C, If[5 < l ≤ 6, ((l - 5) * 180°), 180°], R_2) → C'_9
  275. Segment(R_2, C'_9) → p_6
  276. Rotate(f_5, If[5 < l ≤ 6, ((-(l - 5)) * 180°), -180°], S_2) → q_6
  277. Rotate(S_1, If[5 < l ≤ 6, ((-(l - 5)) * 180°), -180°], S_2) → C_3
  278. Segment(S_2, C_3) → d_7
  279. Rotate(J_2, 3.141592653589793, S_2) → D_3
  280. Line(N_1, D_3) → g_7

AI 绘图提示语参考

绘制一个正八边形,首先以点D为圆心画一个圆,与通过点D的射线相交得到点U和V,以此类推,利用点和射线构造其他点,连接这些点形成正八边形。注意角度和对称性。
以上提示语为根据原始绘图命令整理而来,仅供参考。