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If Pythagoras Had Regular Heptagons: Exploring Pythagorean Relations in Heptagon

If Pythagoras Had Regular Heptagons: Exploring Pythagorean Relations in Heptagon
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2023-03-21 10:44
PolygonAngleVectorTrigonometryGeometric Construction
This example demonstrates the exploration of finding geometric relationships related to the Pythagorean theorem within regular heptagons. The core mathematical concepts include: properties of regular heptagons, extensions of the Pythagorean theorem in special polygons, construction and area ratios of isosceles triangles, and applications of trigonometric functions in polygon calculations. Construction details: Use Polygon command to construct regular heptagons poly1 and poly2, find centroid with Centroid, create midpoints with Midpoint, draw line segments with Segment, use Circle and Line for auxiliary constructions, apply OrthogonalLine for perpendiculars, Intersect to find intersections, Rotate for rotation transformations, Vector for translations, and Angle to measure angles. Suitable for upper secondary school and introductory college geometry courses, exploring the deep connections between polygons and the Pythagorean theorem, and understanding the role of trigonometric functions in geometric proofs.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a regular heptagon. On one of its sides, construct an isosceles triangle sharing that side as base. Using the heptagon's center as pivot, rotate the triangle and heptagon by multiples of one-seventh of a full turn to create symmetric copies. Connect vertices of copies to the center with auxiliary segments, and mark midpoints of these segments. Add auxiliary circles and perpendicular lines to determine key intersection points. Finally, color-code the original and rotated copies, and label all angles.
The prompt above was compiled from the original drawing commands and is for reference only.