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GeoGebra Construction Inspired by Alcázar de Sevilla: Midpoints, Mirrors, and Or

GeoGebra Construction Inspired by Alcázar de Sevilla: Midpoints, Mirrors, and Or
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2023-03-22 22:45
CirclePolygonSolid GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to recreate a visually structured, symmetry-driven geometry inspired by a “site layout” using classic construction steps. It highlights key ideas such as midpoint construction, mirror (reflection) operations, vector-based representation, and orthogonal (perpendicular) line building, alongside circle and polygon constraints. A small amount of conditional logic (If) and selection via Min is used to guide which geometric configuration is active. Construction is carried out with GeoGebra commands including Midpoint, Mirror, Line, OrthogonalLine, Point, Vector, Length, If, Polygon, Intersect, Min, Circle, and Translate, producing corresponding point sets (including mirrored/translated variants) and the final polygon/circle framework. Suitable for advanced secondary students, geometry construction lessons, and introductory math modeling where transformation geometry and constraint-based drafting are emphasized.

GeoGebra Commands & Parameters

  1. Midpoint(D, C) → E
  2. Mirror(E, C) → E'
  3. Midpoint(E', C) → H
  4. Line(A, B) → b
  5. OrthogonalLine(C, b) → d
  6. Line(H, d) → j
  7. Mirror(H, j) → H'_1
  8. Line(E', d) → g
  9. Mirror(H'_1, g) → Z
  10. Point(d) → F
  11. Mirror(F, j) → F'_2
  12. Midpoint(H, F) → K
  13. Mirror(C, K) → C'_1
  14. Mirror(H, C'_1) → H'_2
  15. Mirror(C'_1, j) → W
  16. Mirror(W, g) → W'
  17. Mirror(F'_2, W') → M_1
  18. Mirror(W', M_1) → W''
  19. Mirror(Z, W') → Z'
  20. Mirror(H'_2, Z') → L_1
  21. Vector(L_1, W'') → q
  22. Length(q) → l1
  23. If(a < 2.3, 0, If[a < 3.3, a - 2.3, 1]) → d3
  24. If(a < 1.2, 0, If[a < 2.2, a - 1.2, 1]) → d2
  25. If(a < 3.4, 0, If[a < 4.4, a - 3.4, 1]) → d4
  26. If(a < 4.5, 0, If[a < 5.5, a - 4.5, 1]) → d5
  27. If(a < 5.6, 0, If[a < 6.6, a - 5.6, 1]) → d6
  28. If(a < 6.7, 0, If[a < 7.7, a - 6.7, 1]) → d7
  29. If(a < 7.8, 0, If[a < 8.8, a - 7.8, 1]) → d8
  30. Mirror(D, C) → D'
  31. Mirror(F, C) → F'
  32. Mirror(C, F) → C'
  33. Line(E, d) → e
  34. Line(D, d) → f
  35. Line(D', d) → h
  36. Midpoint(D', E') → G
  37. Midpoint(C, E) → I
  38. Midpoint(E, D) → J
  39. Line(G, d) → i
  40. Line(I, d) → k
  41. Line(J, d) → l
  42. OrthogonalLine(F, d) → m
  43. Polygon(H, C, F, C'_1) → poly1, h_1, c_1, f_1, c'_1
  44. Point(c_1) → L
  45. Point(h_1) → M
  46. Point(c'_1) → N
  47. OrthogonalLine(M, f_1) → n
  48. Intersect(f_1, n) → P
  49. Midpoint(c_1) → Q
  50. Mirror(L, Q) → L'
  51. Polygon(L, M, N, P, L') → poly2, l_1, m_1, n_1, p, l'
  52. Mirror(H, d) → H'
  53. Mirror(C, d) → C'_2
  54. Mirror(F, d) → F'_1
  55. Mirror(C'_1, d) → R
  56. Polygon(H', C'_2, F'_1, R) → poly1', h', c'_2, f'_1, r
  57. Mirror(L, d) → L'_1
  58. Mirror(M, d) → M'
  59. Mirror(N, d) → N'
  60. Mirror(P, d) → P'
  61. Mirror(L', d) → L''
  62. Polygon(L'_1, M', N', P', L'') → poly2', l'_1, m', n', p', l''
  63. Mirror(L'_1, k) → S
  64. Mirror(M', k) → M''
  65. Mirror(N', k) → N''
  66. Mirror(P', k) → P''
  67. Mirror(L'', k) → L'''
  68. Mirror(H', k) → H''
  69. Mirror(C'_2, k) → T
  70. Mirror(F'_1, k) → U
  71. Mirror(R, k) → R'
  72. Polygon(H'', T, U, R') → poly1'', h'', t, u, r'
  73. Mirror(H'', e) → H'''
  74. Mirror(T, e) → T'
  75. Mirror(U, e) → U'
  76. Mirror(R', e) → R''
  77. Polygon(H''', T', U', R'') → poly1''', h''', t', u', r''
  78. Mirror(S, e) → S'
  79. Mirror(M'', e) → M'''
  80. Mirror(N'', e) → N'''
  81. Mirror(P'', e) → P'''
  82. Mirror(L''', e) → V
  83. Mirror(C, j) → C'_3
  84. Polygon(H'_1, C'_3, F'_2, W) → poly1'_1, h'_1, c'_3, f'_2, w
  85. Mirror(L, j) → L'_2
  86. Mirror(M, j) → M'_1
  87. Mirror(N, j) → N'_1
  88. Mirror(P, j) → P'_1
  89. Mirror(L', j) → L''_1
  90. Polygon(L'_2, M'_1, N'_1, P'_1, L''_1) → poly2'_1, l'_2, m'_1, n'_1, p'_1, l''_1
  91. Mirror(C'_3, g) → A_1
  92. Mirror(F'_2, g) → B_1
  93. Polygon(Z, A_1, B_1, W') → poly3, z, a_1, b_1, w'
  94. Mirror(L'_2, g) → C_1
  95. Mirror(M'_1, g) → D_1
  96. Mirror(N'_1, g) → E_1
  97. Mirror(P'_1, g) → F_1
  98. Mirror(L''_1, g) → G_1
  99. Polygon(C_1, D_1, E_1, F_1, G_1) → poly4, c_2, d_1, e_1, f_2, g_1
  100. Mirror(L'_1, k) → O
  101. Mirror(M', k) → M''_1
  102. Mirror(N', k) → N''_1
  103. Mirror(P', k) → P''_1
  104. Mirror(L'', k) → L'''_1
  105. Polygon(O, M''_1, N''_1, P''_1, L'''_1) → poly2'', o, m''_1, n''_1, p''_1, l'''_1
  106. Mirror(O, e) → O'
  107. Mirror(M''_1, e) → H_1
  108. Mirror(N''_1, e) → I_1
  109. Mirror(P''_1, e) → J_1
  110. Mirror(L'''_1, e) → K_1
  111. Polygon(O', H_1, I_1, J_1, K_1) → poly2''', o', h_2, i_1, j_1, k_1
  112. Midpoint(Z', L_1) → N_1
  113. Vector(L_1, Z') → s
  114. Length(s) → l2
  115. Min(l1, ((a - 3) * l1)) → e1
  116. Vector(W', W'') → u_1
  117. Vector(W', Z') → v_1
  118. Min(l1, ((a - 6) * l1)) → e2
  119. OrthogonalLine(O_1, b) → p_1
  120. OrthogonalLine(O_1, p_1) → q_1
  121. Circle(O_1, e1) → r_1
  122. Circle(O_1, e2) → s_1
  123. Intersect(r_1, p_1, 2) → P_1
  124. Vector(O_1, P_1) → w_1
  125. Intersect(s_1, p_1, 1) → Q_1
  126. Vector(O_1, Q_1) → z_1
  127. Translate(H, w_1) → H'_3
  128. Translate(C, w_1) → C'_4
  129. Translate(F, w_1) → F'_3
  130. Translate(C'_1, w_1) → R_1
  131. Polygon(H'_3, C'_4, F'_3, R_1) → poly1'_2, h'_3, c'_4, f'_3, r_2
  132. Translate(H', w_1) → H''_1
  133. Translate(C'_2, w_1) → S_1
  134. Translate(F'_1, w_1) → T_1
  135. Translate(R, w_1) → R'_1
  136. Polygon(H''_1, S_1, T_1, R'_1) → poly1''_1, h''_1, s_2, t_1, r'_1
  137. Translate(L, w_1) → L'_3
  138. Translate(M, w_1) → M'_2
  139. Translate(N, w_1) → N'_2
  140. Translate(P, w_1) → P'_2
  141. Translate(L', w_1) → L''_2
  142. Polygon(L'_3, M'_2, N'_2, P'_2, L''_2) → poly2'_2, l'_3, m'_2, n'_2, p'_2, l''_2
  143. Translate(L'_1, w_1) → U_1
  144. Translate(M', w_1) → M''_2
  145. Translate(N', w_1) → N''_2
  146. Translate(P', w_1) → P''_2
  147. Translate(L'', w_1) → L'''_2
  148. Polygon(U_1, M''_2, N''_2, P''_2, L'''_2) → poly2''_1, u_2, m''_2, n''_2, p''_2, l'''_2
  149. Translate(H, z_1) → H'_4
  150. Translate(C, z_1) → C'_5
  151. Translate(F, z_1) → F'_4
  152. Translate(C'_1, z_1) → V_1
  153. Polygon(H'_4, C'_5, F'_4, V_1) → poly1'_3, h'_4, c'_5, f'_4, v_2
  154. Translate(L, z_1) → L'_4
  155. Translate(M, z_1) → M'_3
  156. Translate(N, z_1) → N'_3
  157. Translate(P, z_1) → P'_3
  158. Translate(L', z_1) → L''_3
  159. Polygon(L'_4, M'_3, N'_3, P'_3, L''_3) → poly2'_3, l'_4, m'_3, n'_3, p'_3, l''_3
  160. Translate(p_1, z_1) → a_2
  161. Translate(H', z_1) → H''_2
  162. Translate(C'_2, z_1) → W_1
  163. Translate(F'_1, z_1) → Z_1
  164. Translate(R, z_1) → R'_2
  165. Polygon(H''_2, W_1, Z_1, R'_2) → poly1''_2, h''_2, w_2, z_2, r'_2
  166. Translate(L'_1, z_1) → A_2
  167. Translate(M', z_1) → M''_3
  168. Translate(N', z_1) → N''_3
  169. Translate(P', z_1) → P''_3
  170. Translate(L'', z_1) → L'''_3
  171. Polygon(A_2, M''_3, N''_3, P''_3, L'''_3) → poly2''_2, a_3, m''_3, n''_3, p''_3, l'''_3
  172. Min(l2, ((a - 4.5) * l2)) → e3
  173. Circle(O_1, e3) → d_2
  174. Min(l2, ((a - 6.5) * l2)) → e4
  175. Intersect(d_2, q_1, 2) → B_2
  176. Vector(O_1, B_2) → b_2
  177. Intersect(d_2, q_1, 1) → D_2
  178. Vector(O_1, D_2) → i_2
  179. Translate(H'_3, b_2) → E_2
  180. Translate(C'_4, b_2) → F_2
  181. Translate(F'_3, b_2) → G_2
  182. Translate(R_1, b_2) → H_2
  183. Polygon(E_2, F_2, G_2, H_2) → poly5, e_3, f_3, g_3, h_3
  184. Translate(H''_1, b_2) → I_2
  185. Translate(S_1, b_2) → J_2
  186. Translate(T_1, b_2) → K_2
  187. Translate(R'_1, b_2) → L_2
  188. Polygon(I_2, J_2, K_2, L_2) → poly6, i_3, j_2, k_2, l_2
  189. Translate(L'_3, b_2) → M_2
  190. Translate(M'_2, b_2) → N_2
  191. Translate(N'_2, b_2) → O_2
  192. Translate(P'_2, b_2) → P_2
  193. Translate(L''_2, b_2) → Q_2
  194. Polygon(M_2, N_2, O_2, P_2, Q_2) → poly7, m_2, n_2, o_2, p_2, q_2
  195. Translate(U_1, b_2) → R_2
  196. Translate(M''_2, b_2) → S_2
  197. Translate(N''_2, b_2) → T_2
  198. Translate(P''_2, b_2) → U_2
  199. Translate(L'''_2, b_2) → V_2
  200. Polygon(R_2, S_2, T_2, U_2, V_2) → poly8, r_3, s_3, t_2, u_3, v_3
  201. Translate(H'_3, i_2) → W_2
  202. Translate(C'_4, i_2) → Z_2
  203. Translate(F'_3, i_2) → A_3
  204. Translate(R_1, i_2) → B_3
  205. Polygon(W_2, Z_2, A_3, B_3) → poly9, w_3, z_3, a_4, b_3
  206. Translate(H''_1, i_2) → C_3
  207. Translate(S_1, i_2) → D_3
  208. Translate(T_1, i_2) → E_3
  209. Translate(R'_1, i_2) → F_3
  210. Polygon(C_3, D_3, E_3, F_3) → poly10, c_3, d_3, e_4, f_4
  211. Translate(L'_3, i_2) → G_3
  212. Translate(M'_2, i_2) → H_3
  213. Translate(N'_2, i_2) → I_3
  214. Translate(P'_2, i_2) → J_3
  215. Translate(L''_2, i_2) → K_3
  216. Polygon(G_3, H_3, I_3, J_3, K_3) → poly11, g_4, h_4, i_4, j_3, k_3
  217. Translate(U_1, i_2) → L_3
  218. Translate(M''_2, i_2) → M_3
  219. Translate(N''_2, i_2) → N_3
  220. Translate(P''_2, i_2) → O_3
  221. Translate(L'''_2, i_2) → P_3
  222. Polygon(L_3, M_3, N_3, O_3, P_3) → poly12, l_3, m_3, n_3, o_3, p_3
  223. OrthogonalLine(C_2, b) → e_2
  224. OrthogonalLine(C_2, e_2) → g_2
  225. Circle(C_2, e4) → q_3
  226. Intersect(q_3, g_2, 2) → Q_3
  227. Vector(C_2, Q_3) → t_3
  228. Intersect(q_3, g_2, 1) → R_3
  229. Vector(C_2, R_3) → u_4
  230. Translate(g_2, t_3) → b_4
  231. Translate(H'_4, t_3) → S_3
  232. Translate(C'_5, t_3) → T_3
  233. Translate(F'_4, t_3) → U_3
  234. Translate(V_1, t_3) → V_3
  235. Polygon(S_3, T_3, U_3, V_3) → poly13, s_4, t_4, u_5, v_4
  236. Translate(L'_4, t_3) → W_3
  237. Translate(M'_3, t_3) → Z_3
  238. Translate(N'_3, t_3) → A_4
  239. Translate(P'_3, t_3) → B_4
  240. Translate(L''_3, t_3) → C_4
  241. Polygon(W_3, Z_3, A_4, B_4, C_4) → poly14, w_4, z_4, a_5, b_5, c_4
  242. Translate(H''_2, u_4) → D_4
  243. Translate(W_1, u_4) → E_4
  244. Translate(Z_1, u_4) → F_4
  245. Translate(R'_2, u_4) → G_4
  246. Polygon(D_4, E_4, F_4, G_4) → poly15, d_4, e_5, f_5, g_5
  247. Translate(A_2, u_4) → H_4
  248. Translate(M''_3, u_4) → I_4
  249. Translate(N''_3, u_4) → J_4
  250. Translate(P''_3, u_4) → K_4
  251. Translate(L'''_3, u_4) → L_4
  252. Polygon(H_4, I_4, J_4, K_4, L_4) → poly16, h_5, i_5, j_4, k_4, l_4
  253. Translate(H'_4, u_4) → M_4
  254. Translate(C'_5, u_4) → N_4
  255. Translate(F'_4, u_4) → O_4
  256. Translate(V_1, u_4) → P_4
  257. Polygon(M_4, N_4, O_4, P_4) → poly17, m_4, n_4, o_4, p_4
  258. Translate(L'_4, u_4) → Q_4
  259. Translate(M'_3, u_4) → R_4
  260. Translate(N'_3, u_4) → S_4
  261. Translate(P'_3, u_4) → T_4
  262. Translate(L''_3, u_4) → U_4
  263. Polygon(Q_4, R_4, S_4, T_4, U_4) → poly18, q_4, r_4, s_5, t_5, u_6
  264. Translate(H''_2, t_3) → V_4
  265. Translate(W_1, t_3) → W_4
  266. Translate(Z_1, t_3) → Z_4
  267. Translate(R'_2, t_3) → A_5
  268. Polygon(V_4, W_4, Z_4, A_5) → poly19, v_5, w_5, z_5, a_6
  269. Translate(A_2, t_3) → B_5
  270. Translate(M''_3, t_3) → C_5
  271. Translate(N''_3, t_3) → D_5
  272. Translate(P''_3, t_3) → E_5
  273. Translate(L'''_3, t_3) → F_5
  274. Polygon(B_5, C_5, D_5, E_5, F_5) → poly20, b_6, c_5, d_5, e_6, f_6

AI Drawing Prompt Reference

Create a geometric composition based on midpoints, mirrors, and perpendicular lines. Start with two base points C and D, construct their midpoint E, then mirror E about C to get E', and find the midpoint of E' and C to obtain H. Draw a line through points A and B, then create a perpendicular line d through C. Through H, draw a line j parallel to d, and mirror H about j to get H'_1. Continue constructing key points like F, W, Z through reflections and lines. Finally, draw a quadrilateral with vertices H, C, F, and C'_1. The entire figure demonstrates symmetry and transformation geometry.
The prompt above was compiled from the original drawing commands and is for reference only.