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如果毕达哥拉斯遇见正则七边形：七边形中的勾股关系探索

如果毕达哥拉斯遇见正则七边形：七边形中的勾股关系探索
几何画板及绘图内容加载中，请稍等……
2023-03-21 10:44
多边形角向量三角函数几何作图
本范例演示了在正则七边形（heptagon）中寻找与勾股定理相关几何关系的过程。核心数学概念包括：正则七边形的性质、勾股定理在特殊多边形中的推广、等腰三角形的构造与面积比、三角函数在多边形计算中的应用。作图要点：使用Polygon命令构造正则七边形poly1和poly2，通过Centroid求重心、Midpoint作中点、Segment连线、Circle和Line作辅助线，利用OrthogonalLine作垂线、Intersect找交点、Rotate旋转、Vector平移变换、Angle测量角度，构建完整的几何推理体系。适合中学高年级及大学初等几何课程，可用于探索多边形与勾股定理的深层联系，理解三角函数在几何证明中的作用。

GeoGebra 命令及参数

  1. Polygon(B, A, 7) → poly1, i, H, I, J, j, k, l, m, n, p, F, G
  2. Midpoint(n) → W
  3. Centroid(poly1) → U
  4. Segment(U, W) → m_1
  5. Circle(A, B) → d
  6. Circle(A, 1) → c
  7. Line(A, B) → f
  8. OrthogonalLine(A, f) → g
  9. Intersect(c, g, 2) → C
  10. Ray(A, C) → h
  11. Intersect(d, h, 1) → D
  12. Point(Ray[A, D]) → E
  13. Polygon(E, A, B) → t1, b_1, e_1, a_1
  14. InteriorAngles(t1) → α, β, γ
  15. Rotate(W, γ, I) → W'
  16. Line(I, W') → n_1
  17. Intersect(n_1, m_1) → E_1
  18. Dilate(E_1, If[7 < o ≤ 8, 8 - o, 0], E) → R_3
  19. Polygon(A, E, 7) → poly2, q, M, N, O, r, s, t, a, b, e, K, L
  20. Polygon(E, B, 7) → poly3, f_1, R, S, T, g_1, h_1, i_1, j_1, k_1, l_1, P, Q
  21. Midpoint(p) → V
  22. Midpoint(m) → Z
  23. Midpoint(l) → A_1
  24. Midpoint(k) → B_1
  25. Midpoint(j) → C_1
  26. Midpoint(i) → D_1
  27. Polygon(E_1, W, I) → t2, i_2, p_1, w
  28. Polygon(U, E_1, I, Z) → q1, u, q_1, i_3, z_1
  29. Rotate(U, (2 * pi / 7), U) → U'
  30. Rotate(E_1, (2 * pi / 7), U) → F_1
  31. Rotate(I, (2 * pi / 7), U) → I'
  32. Rotate(Z, (2 * pi / 7), U) → Z'
  33. Polygon(U', F_1, I', Z') → q1', u', r_1, i', z'
  34. Rotate(E_1, (2 * pi / 7), U) → G_1
  35. Rotate(W, (2 * pi / 7), U) → W'_{1}
  36. Rotate(I, (2 * pi / 7), U) → I'_{1}
  37. Polygon(G_1, W'_{1}, I'_{1}) → t2', i'_{1}, s_1, w'_{1}
  38. Rotate(E_1, (4 * pi / 7), U) → H_1
  39. Rotate(W, (4 * pi / 7), U) → W'_{2}
  40. Rotate(I, (4 * pi / 7), U) → I'_{2}
  41. Polygon(H_1, W'_{2}, I'_{2}) → t2'_{1}, i'_{2}, t_1, w'_{2}
  42. Rotate(U, (4 * pi / 7), U) → U'_{1}
  43. Rotate(E_1, (4 * pi / 7), U) → I_1
  44. Rotate(I, (4 * pi / 7), U) → I'_{3}
  45. Rotate(Z, (4 * pi / 7), U) → Z'_{1}
  46. Polygon(U'_{1}, I_1, I'_{3}, Z'_{1}) → q1'_{1}, u'_{1}, c_1, i'_{3}, z'_{1}
  47. Rotate(U, (6 * pi / 7), U) → U'_{2}
  48. Rotate(E_1, (6 * pi / 7), U) → J_1
  49. Rotate(I, (6 * pi / 7), U) → I'_{4}
  50. Rotate(Z, (6 * pi / 7), U) → Z'_{2}
  51. Polygon(U'_{2}, J_1, I'_{4}, Z'_{2}) → q1'_{2}, u'_{2}, d_1, i'_{4}, z'_{2}
  52. Rotate(E_1, (6 * pi / 7), U) → K_1
  53. Rotate(W, (6 * pi / 7), U) → W'_{3}
  54. Rotate(I, (6 * pi / 7), U) → I'_{5}
  55. Polygon(K_1, W'_{3}, I'_{5}) → t2'_{2}, i'_{5}, f_2, w'_{3}
  56. Rotate(E_1, (8 * pi / 7), U) → L_1
  57. Rotate(W, (8 * pi / 7), U) → W'_{4}
  58. Rotate(I, (8 * pi / 7), U) → I'_{6}
  59. Polygon(L_1, W'_{4}, I'_{6}) → t2'_{3}, i'_{6}, g_2, w'_{4}
  60. Rotate(U, (8 * pi / 7), U) → U'_{3}
  61. Rotate(E_1, (8 * pi / 7), U) → M_1
  62. Rotate(I, (8 * pi / 7), U) → I'_{7}
  63. Rotate(Z, (8 * pi / 7), U) → Z'_{3}
  64. Polygon(U'_{3}, M_1, I'_{7}, Z'_{3}) → q1'_{3}, u'_{3}, h_2, i'_{7}, z'_{3}
  65. Rotate(U, (10 * pi / 7), U) → U'_{4}
  66. Rotate(E_1, (10 * pi / 7), U) → N_1
  67. Rotate(I, (10 * pi / 7), U) → I'_{8}
  68. Rotate(Z, (10 * pi / 7), U) → Z'_{4}
  69. Polygon(U'_{4}, N_1, I'_{8}, Z'_{4}) → q1'_{4}, u'_{4}, j_2, i'_{8}, z'_{4}
  70. Rotate(E_1, (10 * pi / 7), U) → O_1
  71. Rotate(W, (10 * pi / 7), U) → W'_{5}
  72. Rotate(I, (10 * pi / 7), U) → I'_{9}
  73. Polygon(O_1, W'_{5}, I'_{9}) → t2'_{4}, i'_{9}, o_1, w'_{5}
  74. Rotate(E_1, (12 * pi / 7), U) → P_1
  75. Rotate(W, (12 * pi / 7), U) → W'_{6}
  76. Rotate(I, (12 * pi / 7), U) → I'_{10}
  77. Polygon(P_1, W'_{6}, I'_{10}) → t2'_{5}, i'_{10}, k_2, w'_{6}
  78. Rotate(U, (12 * pi / 7), U) → U'_{5}
  79. Rotate(E_1, (12 * pi / 7), U) → Q_1
  80. Rotate(I, (12 * pi / 7), U) → I'_{11}
  81. Rotate(Z, (12 * pi / 7), U) → Z'_{5}
  82. Polygon(U'_{5}, Q_1, I'_{11}, Z'_{5}) → q1'_{5}, u'_{5}, l_2, i'_{11}, z'_{5}
  83. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], R) → R_1
  84. Vector(A, R_1) → v
  85. Translate(U'_{2}, v) → S_1
  86. Translate(J_1, v) → T_1
  87. Translate(I'_{4}, v) → U_1
  88. Translate(Z'_{2}, v) → V_1
  89. Polygon(S_1, T_1, U_1, V_1) → q2, m_2, n_2, u_1, v_1
  90. Translate(K_1, v) → W_1
  91. Translate(W'_{3}, v) → Z_1
  92. Translate(I'_{5}, v) → A_2
  93. Angle(S, R, T_1) → δ
  94. Rotate(S_1, If[0 < o ≤ 1, ((-o) * δ), (-δ)], R_1) → B_2
  95. Rotate(T_1, If[0 < o ≤ 1, ((-o) * δ), (-δ)], R_1) → C_2
  96. Rotate(U_1, If[0 < o ≤ 1, ((-o) * δ), (-δ)], R_1) → D_2
  97. Rotate(V_1, If[0 < o ≤ 1, ((-o) * δ), (-δ)], R_1) → E_2
  98. Polygon(B_2, C_2, D_2, E_2) → q2', b_2, c_2, d_2, e_2
  99. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], S) → A'
  100. Vector(A, A') → w_1
  101. Translate(K_1, w_1) → F_2
  102. Translate(W'_{3}, w_1) → G_2
  103. Translate(I'_{5}, w_1) → H_2
  104. Polygon(F_2, G_2, H_2) → t3, p_2, q_2, r_2
  105. Angle(F_2, A', C_2) → ε
  106. Translate(J_1, w_1) → I_2
  107. Rotate(F_2, If[1 < o ≤ 2, ((o - 1) * ε), ε], A') → J_2
  108. Rotate(G_2, If[1 < o ≤ 2, ((o - 1) * ε), ε], A') → K_2
  109. Rotate(H_2, If[1 < o ≤ 2, ((o - 1) * ε), ε], A') → L_2
  110. Polygon(J_2, K_2, L_2) → t3', s_2, t_2, a_2
  111. Dilate(B, If[2 < o ≤ 3, 3 - o, 0], S) → B'
  112. Vector(B, B') → u_2
  113. Translate(U'_{1}, u_2) → M_2
  114. Translate(I_1, u_2) → N_2
  115. Translate(I'_{3}, u_2) → O_2
  116. Translate(Z'_{1}, u_2) → P_2
  117. Polygon(M_2, N_2, O_2, P_2) → q3, f_3, g_3, o_2, h_3
  118. Translate(V, Vector[B, S]) → V'
  119. Angle(V', S, K_2) → ζ
  120. Rotate(M_2, If[2 < o ≤ 3, ((o - 2) * ζ), ζ], B') → Q_2
  121. Rotate(N_2, If[2 < o ≤ 3, ((o - 2) * ζ), ζ], B') → R_2
  122. Rotate(O_2, If[2 < o ≤ 3, ((o - 2) * ζ), ζ], B') → S_2
  123. Rotate(P_2, If[2 < o ≤ 3, ((o - 2) * ζ), ζ], B') → T_2
  124. Polygon(Q_2, R_2, S_2, T_2) → q3', j_3, k_3, l_3, m_3
  125. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], T) → B'_{1}
  126. Vector(B, B'_{1}) → v_2
  127. Translate(H_1, v_2) → U_2
  128. Translate(W'_{2}, v_2) → V_2
  129. Translate(I'_{2}, v_2) → W_2
  130. Polygon(U_2, V_2, W_2) → t4, w_2, n_3, p_3
  131. Translate(H_1, Vector[B, T]) → Z_2
  132. Angle(Z_2, T, R_2) → η
  133. Rotate(U_2, If[3 < o ≤ 4, ((o - 3) * η), η], B'_{1}) → A_3
  134. Rotate(V_2, If[3 < o ≤ 4, ((o - 3) * η), η], B'_{1}) → B_3
  135. Rotate(W_2, If[3 < o ≤ 4, ((o - 3) * η), η], B'_{1}) → C_3
  136. Polygon(A_3, B_3, C_3) → t4', c_3, a_3, b_3
  137. Dilate(J, If[4 < o ≤ 5, 5 - o, 0], T) → J'
  138. Vector(J, J') → u_3
  139. Translate(U', u_3) → U''
  140. Translate(F_1, u_3) → D_3
  141. Translate(I', u_3) → I''
  142. Translate(Z', u_3) → Z''
  143. Polygon(U'', D_3, I'', Z'') → q1'', u'', d_3, i'', z''
  144. Translate(W, Vector[J, T]) → W'_{7}
  145. Angle(W'_{7}, T, B_3) → θ
  146. Rotate(U'', If[4 < o ≤ 5, ((-(o - 4)) * ((2 * pi) - θ)), (-((2 * pi) - θ))], J') → U'''
  147. Rotate(D_3, If[4 < o ≤ 5, ((-(o - 4)) * ((2 * pi) - θ)), (-((2 * pi) - θ))], J') → E_3
  148. Rotate(I'', If[4 < o ≤ 5, ((-(o - 4)) * ((2 * pi) - θ)), (-((2 * pi) - θ))], J') → I'''
  149. Rotate(Z'', If[4 < o ≤ 5, ((-(o - 4)) * ((2 * pi) - θ)), (-((2 * pi) - θ))], J') → Z'''
  150. Polygon(U''', E_3, I''', Z''') → q1''', u''', e_3, i''', z'''
  151. Dilate(F_1, If[5 < o ≤ 6, 6 - o, 0], E_3) → F_3
  152. Vector(F_1, F_3) → v_3
  153. Translate(G_1, v_3) → G_3
  154. Translate(W'_{1}, v_3) → H_3
  155. Translate(I'_{1}, v_3) → I_3
  156. Polygon(G_3, H_3, I_3) → t2'', q_3, r_3, s_3
  157. Translate(J, Vector[F_1, E_3]) → J'_{1}
  158. Angle(M, L, K) → ι
  159. Angle(E, E_3, J'_{1}) → κ
  160. Rotate(G_3, If[5 < o ≤ 6, ((-(o - 5)) * κ), (-κ)], F_3) → J_3
  161. Rotate(H_3, If[5 < o ≤ 6, ((-(o - 5)) * κ), (-κ)], F_3) → K_3
  162. Rotate(I_3, If[5 < o ≤ 6, ((-(o - 5)) * κ), (-κ)], F_3) → L_3
  163. Polygon(J_3, K_3, L_3) → t2''', t_3, f_4, g_4
  164. Dilate(I, If[6 < o ≤ 7, 7 - o, 0], E) → I'_{12}
  165. Vector(I, I'_{12}) → w_3
  166. Translate(U, w_3) → U'_{6}
  167. Translate(E_1, w_3) → M_3
  168. Translate(I, w_3) → I'_{13}
  169. Translate(Z, w_3) → Z'_{6}
  170. Polygon(U'_{6}, M_3, I'_{13}, Z'_{6}) → q1'_{6}, u'_{6}, h_4, i'_{13}, z'_{6}
  171. Translate(Z, Vector[I, E]) → Z'_{7}
  172. Angle(Z'_{7}, E, K_3) → λ
  173. Rotate(U'_{6}, If[6 < o ≤ 7, ((-(o - 6)) * ((2 * pi) - λ)), λ - (2 * pi)], I'_{12}) → N_3
  174. Rotate(M_3, If[6 < o ≤ 7, ((-(o - 6)) * ((2 * pi) - λ)), λ - (2 * pi)], I'_{12}) → O_3
  175. Rotate(I'_{13}, If[6 < o ≤ 7, ((-(o - 6)) * ((2 * pi) - λ)), λ - (2 * pi)], I'_{12}) → P_3
  176. Rotate(Z'_{6}, If[6 < o ≤ 7, ((-(o - 6)) * ((2 * pi) - λ)), λ - (2 * pi)], I'_{12}) → Q_3
  177. Polygon(N_3, O_3, P_3, Q_3) → q4, i_4, o_3, j_4, k_4
  178. Vector(E_1, R_3) → u_4
  179. Translate(I, u_4) → I'_{14}
  180. Translate(E_1, Vector[W, E]) → I'_{15}
  181. Translate(I, Vector[W, E]) → I'_{16}
  182. Translate(E_1, u_4) → T_3
  183. Translate(I, u_4) → I'_{17}
  184. Translate(I, Vector[E_1, E]) → I'_{18}
  185. Angle(M, L, K) → ν
  186. Angle(A, E, I'_{18}) → ξ
  187. Rotate(T_3, If[7 < o ≤ 8, ((-(o - 7)) * ξ), (-ξ)], R_3) → U_3
  188. Rotate(I'_{17}, If[7 < o ≤ 8, ((-(o - 7)) * ξ), (-ξ)], R_3) → W_3
  189. Translate(H, Vector[A_1, O_3]) → H'
  190. Translate(U, Vector[A_1, O_3]) → U'_{7}
  191. Translate(P_1, Vector[A_1, O_3]) → D_4
  192. Dilate(I, If[7 < o ≤ 8, 8 - o, 0], B) → I'_{19}
  193. Vector(I, I'_{19}) → v_4
  194. Translate(E_1, v_4) → S_3
  195. Translate(W, v_4) → W'_{8}
  196. Translate(I, v_4) → I'_{20}
  197. Polygon(S_3, W'_{8}, I'_{20}) → t2'_{6}, i'_{20}, l_4, w'_{8}
  198. Translate(E_1, Vector[I, B]) → V_3
  199. Angle(V_3, B, O_3) → μ
  200. Rotate(S_3, If[7 < o ≤ 8, ((o - 7) * μ), μ], I'_{19}) → Z_3
  201. Rotate(W'_{8}, If[7 < o ≤ 8, ((o - 7) * μ), μ], I'_{19}) → A_4
  202. Rotate(I'_{20}, If[7 < o ≤ 8, ((o - 7) * μ), μ], I'_{19}) → B_4
  203. Polygon(Z_3, A_4, B_4) → t5, b_4, z_3, a_4
  204. Dilate(A_1, If[8 < o ≤ 9, 9 - o, 0], A_4) → H'_{1}
  205. Vector(A_1, H'_{1}) → w_4
  206. Vector(H, B) → c_4
  207. Translate(U'_{5}, w_4) → C_4
  208. Translate(Q_1, w_4) → E_4
  209. Translate(I'_{11}, w_4) → F_4
  210. Translate(Z'_{5}, w_4) → G_4
  211. Polygon(C_4, E_4, F_4, G_4) → q5, m_4, e_4, n_4, p_4
  212. Translate(H, Vector[A_1, A_4]) → H'_{2}
  213. Angle(M, L, K) → ο
  214. Angle(B, A_4, H'_{2}) → ρ
  215. Rotate(C_4, If[8 < o ≤ 9, ((-(o - 8)) * ρ), (-ρ)], H'_{1}) → H_4
  216. Rotate(E_4, If[8 < o ≤ 9, ((-(o - 8)) * ρ), (-ρ)], H'_{1}) → I_4
  217. Rotate(F_4, If[8 < o ≤ 9, ((-(o - 8)) * ρ), (-ρ)], H'_{1}) → J_4
  218. Rotate(G_4, If[8 < o ≤ 9, ((-(o - 8)) * ρ), (-ρ)], H'_{1}) → K_4
  219. Polygon(H_4, I_4, J_4, K_4) → q5', q_4, r_4, s_4, t_4
  220. Dilate(Z, If[9 < o ≤ 10, 10 - o, 0], I_4) → Z'_{8}
  221. Translate(H, Vector[Z, I_4]) → H'_{3}
  222. Dilate(P_1, If[9 < o ≤ 10, 10 - o, 0], I_4) → L_4
  223. Vector(P_1, L_4) → u_5
  224. Translate(P_1, u_5) → M_4
  225. Translate(W'_{6}, u_5) → N_4
  226. Translate(I'_{10}, u_5) → O_4
  227. Polygon(M_4, N_4, O_4) → t6, o_4, d_4, f_5
  228. Translate(Z, Vector[P_1, I_4]) → Z'_{9}
  229. Angle(Z'_{9}, I_4, H_4) → σ
  230. Rotate(M_4, If[9 < o < 10, ((o - 9) * σ), σ], L_4) → P_4
  231. Rotate(N_4, If[9 < o < 10, ((o - 9) * σ), σ], L_4) → Q_4
  232. Rotate(O_4, If[9 < o < 10, ((o - 9) * σ), σ], L_4) → R_4
  233. Polygon(P_4, Q_4, R_4) → t6', g_5, h_5, i_5
  234. Dilate(G, If[10 < o ≤ 11, 11 - o, 0], P) → G'
  235. Vector(G, G') → v_5
  236. Translate(U'_{4}, v_5) → S_4
  237. Translate(N_1, v_5) → T_4
  238. Translate(I'_{8}, v_5) → U_4
  239. Translate(Z'_{4}, v_5) → V_4
  240. Polygon(S_4, T_4, U_4, V_4) → q6, j_5, k_5, l_5, m_5
  241. Translate(B_1, Vector[G, P]) → W_4
  242. Angle(W_4, P, Q_4) → τ
  243. Rotate(U_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → B_5
  244. Rotate(V_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → C_5
  245. Rotate(S_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → Z_4
  246. Rotate(T_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → A_5
  247. Rotate(U_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → D_5
  248. Rotate(V_4, If[10 < o < 11, ((o - 10) * τ), τ], G') → E_5
  249. Polygon(Z_4, A_5, D_5, E_5) → q6', z_4, a_5, d_5, e_5
  250. Dilate(N_1, If[11 < o ≤ 12, 12 - o, 0], A_5) → F_5
  251. Vector(N_1, F_5) → w_5
  252. Translate(O_1, w_5) → G_5
  253. Translate(W'_{5}, w_5) → H_5
  254. Translate(I'_{9}, w_5) → I_5
  255. Polygon(G_5, H_5, I_5) → t7, n_5, p_5, q_5
  256. Translate(G, Vector[N_1, F_5]) → G'_{1}
  257. Translate(G, Vector[N_1, A_5]) → G'_{2}
  258. Angle(K, L, M) → υ
  259. Angle(Q, A_5, G'_{2}) → ϕ
  260. Rotate(G_5, If[11 < o < 12, ((-(o - 11)) * ϕ), (-ϕ)], F_5) → J_5
  261. Rotate(H_5, If[11 < o < 12, ((-(o - 11)) * ϕ), (-ϕ)], F_5) → K_5
  262. Rotate(I_5, If[11 < o < 12, ((-(o - 11)) * ϕ), (-ϕ)], F_5) → L_5
  263. Polygon(J_5, K_5, L_5) → t7', r_5, s_5, t_5
  264. Translate(B_1, Vector[F, K_5]) → M_5
  265. Translate(C_1, Vector[F, K_5]) → U_5
  266. Translate(C_1, Vector[N_1, F_5]) → Q_5
  267. Translate(C_1, Vector[F, Q]) → V_5
  268. Angle(V_5, Q, Q_5) → χ
  269. Dilate(F, If[12 < o ≤ 13, 13 - o, 0], Q) → F'
  270. Vector(F, F') → b_5
  271. Translate(U'_{3}, b_5) → N_5
  272. Translate(M_1, b_5) → O_5
  273. Translate(I'_{7}, b_5) → P_5
  274. Translate(Z'_{3}, b_5) → R_5
  275. Polygon(N_5, O_5, P_5, R_5) → q7, c_5, o_5, f_6, g_6
  276. Translate(C_1, Vector[F, Q]) → S_5
  277. Angle(V_5, Q, K_5) → ψ
  278. Rotate(N_5, If[12 < o ≤ 13, ((o - 12) * ψ), ψ], F') → T_5
  279. Rotate(O_5, If[12 < o ≤ 13, ((o - 12) * ψ), ψ], F') → W_5
  280. Rotate(P_5, If[12 < o ≤ 13, ((o - 12) * ψ), ψ], F') → Z_5
  281. Rotate(R_5, If[12 < o ≤ 13, ((o - 12) * ψ), ψ], F') → A_6
  282. Polygon(T_5, W_5, Z_5, A_6) → q7', h_6, i_6, z_5, a_6
  283. Line(W_5, u'_{3}) → j_6
  284. Point(j_6) → B_6
  285. Angle(B_6, W_5, E_2) → ω
  286. Dilate(B_1, If[13 < o ≤ 14, 14 - o, 0], E_2) → C_6
  287. Vector(B_1, C_6) → u_6
  288. Translate(L_1, u_6) → D_6
  289. Translate(W'_{4}, u_6) → E_6
  290. Translate(I'_{6}, u_6) → F_6
  291. Polygon(D_6, E_6, F_6) → t8, k_6, d_6, e_6
  292. Rotate(D_6, If[13 < o ≤ 14, ((o - 13) * ω), ω], C_6) → G_6
  293. Rotate(E_6, If[13 < o ≤ 14, ((o - 13) * ω), ω], C_6) → H_6
  294. Rotate(F_6, If[13 < o ≤ 14, ((o - 13) * ω), ω], C_6) → I_6
  295. Polygon(G_6, H_6, I_6) → t8', l_6, m_6, n_6
  296. Line(H_4, r) → p_6
  297. Point(p_6) → J_6
  298. Angle(Z_4, H_4, J_6) → α_1
  299. Dilate(E, If[14 < o ≤ 15, 15 - o, 0], H_4) → E'
  300. Vector(E, E') → v_6
  301. Translate(poly2, v_6) → poly2'
  302. Rotate(poly2', If[15 < o ≤ 16, ((-(o - 15)) * α_1), (-α_1)], H_4) → poly2''
  303. Angle(B, A, E) → β_1
  304. Segment(K, L) → q_6
  305. Segment(L, M) → r_6
  306. Segment(M, N) → s_6
  307. Segment(N, O) → t_6
  308. Segment(O, A) → b_6
  309. Segment(E, A) → c_6
  310. Segment(E, K) → f_7
  311. Segment(A, B) → g_7
  312. Segment(B, E) → h_7
  313. Segment(A, F) → i_7
  314. Segment(F, G) → j_7
  315. Segment(G, H) → k_7
  316. Segment(H, I) → l_7
  317. Segment(I, J) → m_7
  318. Segment(B, J) → n_7
  319. Segment(B, P) → p_7
  320. Segment(P, Q) → q_7
  321. Segment(Q, R) → r_7
  322. Segment(R, S) → s_7
  323. Segment(S, T) → t_7
  324. Segment(T, E) → a_7

AI 绘图提示语参考

绘制一个正七边形，以其一条边为底边，在外部构造一个等腰三角形，使该三角形与正七边形共享底边。以正七边形的中心为旋转中心，将三角形和七边形按七分之一圆周的整数倍旋转，生成多个对称副本。连接各副本的顶点与中心，形成辅助线段，并标出各线段的中点。在图形中构造辅助圆和垂线，用于确定关键交点。最后，用不同颜色区分原始图形与旋转后的副本，并标注所有角度。
以上提示语为根据原始绘图命令整理而来，仅供参考。