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3.4.8 Museo de Onda: Azulejo 8 - Geometric Construction and Optimization

3.4.8 Museo de Onda: Azulejo 8 - Geometric Construction and Optimization
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2023-03-22 22:34
Coordinate GeometryOptimizationGeometric Construction
In collections geometriadinamica.es
This example demonstrates geometric construction methods for solving optimization problems. Key mathematical concepts include functions, optimization, rational numbers, and solid geometry. Construction involves using GeoGebra commands like Line, Point, OrthogonalLine, Midpoint, and Mirror to create figures, and applying transformations to achieve desired results. Suitable for high school students or college freshmen learning geometric construction techniques and optimization problem-solving methods.

GeoGebra Commands & Parameters

  1. Line(A, B) → b
  2. Line(G, b) → e
  3. Point(e) → J
  4. OrthogonalLine(J, e) → h
  5. OrthogonalLine(G, e) → f
  6. Point(f) → H
  7. OrthogonalLine(H, f) → g
  8. Intersect(g, h) → E
  9. Midpoint(E, H) → I
  10. Midpoint(I, E) → B_1
  11. Midpoint(G, J) → D
  12. Midpoint(J, E) → F
  13. Midpoint(H, G) → K
  14. Midpoint(D, I) → L
  15. Midpoint(K, L) → M
  16. Midpoint(L, F) → N
  17. Midpoint(F, J) → O
  18. Midpoint(J, D) → P
  19. Midpoint(G, D) → Q
  20. Midpoint(K, G) → R
  21. Midpoint(Q, M) → S
  22. Midpoint(L, D) → T
  23. Midpoint(P, N) → U
  24. Midpoint(F, E) → V
  25. Midpoint(K, H) → W
  26. Midpoint(W, V) → Z
  27. Midpoint(H, I) → A_1
  28. Midpoint(N, B_1) → C_1
  29. Midpoint(Z, W) → D_1
  30. Midpoint(D, U) → E_1
  31. Midpoint(U, J) → F_1
  32. Midpoint(T, Q) → G_1
  33. Midpoint(S, G) → H_1
  34. Midpoint(S, K) → I_1
  35. Midpoint(M, T) → J_1
  36. Midpoint(T, N) → K_1
  37. Midpoint(N, O) → L_1
  38. Midpoint(F, C_1) → M_1
  39. Midpoint(C_1, L) → N_1
  40. Midpoint(L, D_1) → O_1
  41. Midpoint(D_1, K) → P_1
  42. Midpoint(H, D_1) → Q_1
  43. Midpoint(D_1, I) → R_1
  44. Midpoint(I, C_1) → S_1
  45. Midpoint(C_1, E) → T_1
  46. Mirror(F_1, h) → U_1
  47. Mirror(L_1, h) → V_1
  48. Mirror(M_1, h) → W_1
  49. Mirror(T_1, h) → Z_1
  50. Mirror(B_1, h) → A_2
  51. Mirror(L_1, h) → C_2
  52. Line(Q, A_2) → d
  53. Line(F, d) → i
  54. Line(C_1, d) → k
  55. Line(S_1, k) → l
  56. Line(R_1, L_1) → j
  57. Line(D_1, j) → m
  58. Line(M, j) → n
  59. Line(I, j) → p
  60. Line(S, A_1) → t
  61. Line(I, T) → a_1
  62. Line(U, B_1) → b_1
  63. Line(E_1, N_1) → c_1
  64. Line(F_1, M_1) → d_1
  65. Line(G_1, R_1) → e_1
  66. Line(W_1, P_1) → g_1
  67. Line(Q_1, Z_1) → h_1
  68. Point(a_1) → F_2
  69. Mirror(T, m) → T'
  70. Mirror(K_1, m) → I_2
  71. Mirror(F_2, m) → L_2
  72. Mirror(T, m) → T'_1
  73. Mirror(F_2, m) → M_2
  74. Mirror(L, m) → L'
  75. Mirror(T, m) → T'_2
  76. Mirror(U, m) → U'_1
  77. Mirror(T, m) → T'_3
  78. Point(b) → A1
  79. Min(90, ((a - 1) * 90)) → r1
  80. Rotate(F_2, ((-r1))°, T) → K_{11}
  81. Rotate(T, ((-r1))°, T) → T'_7
  82. Rotate(U'_1, ((-r1))°, T) → D_{12}
  83. Rotate(T'_3, ((-r1))°, T) → E_{12}
  84. Rotate(U, ((-r1))°, T) → U'_3
  85. Rotate(T, ((-r1))°, T) → T'_8
  86. Rotate(T, ((-r1))°, T'_8) → T'_9
  87. Rotate(T'_2, ((-r1))°, T) → I_{12}
  88. Rotate(L_2, ((-r1))°, T) → S_{12}
  89. Rotate(T'_1, ((-r1))°, T) → T_{12}
  90. Rotate(M_2, ((-r1))°, T) → H_{13}
  91. Rotate(L', ((-r1))°, T) → L''_2
  92. Rotate(F_2, ((-r1))°, T) → J_{13}
  93. Rotate(L, ((-r1))°, T) → L'_2
  94. Mirror(U'_3, b_1) → A_{14}
  95. Mirror(T'_8, b_1) → B_{14}
  96. Mirror(D_{12}, b_1) → D_{14}
  97. Mirror(E_{12}, b_1) → E_{14}
  98. Mirror(I_{12}, b_1) → L_{15}
  99. Mirror(T'_9, b_1) → O_{15}
  100. Mirror(S_{12}, b_1) → D_{16}
  101. Mirror(T_{12}, b_1) → E_{16}
  102. Mirror(K_{11}, b_1) → G_{16}
  103. Mirror(T'_7, b_1) → H_{16}
  104. Mirror(H_{13}, b_1) → G_{17}
  105. Mirror(L''_2, b_1) → H_{17}
  106. Mirror(J_{13}, b_1) → J_{17}
  107. Mirror(L'_2, b_1) → K_{17}
  108. Midpoint(C_1, E) → K_{20}
  109. Rotate(U'_1, 270°, L') → D_{21}
  110. Rotate(D_{21}, 270°, L') → G_{21}
  111. Rotate(G_{21}, 270°, L') → M_{21}
  112. Line(U'_1, D_{21}) → n_1
  113. Mirror(U'_1, n_1) → V_{21}
  114. Mirror(M_{21}, n_1) → D_{22}
  115. Mirror(D_{21}, n_1) → M_{24}
  116. Mirror(G_{21}, n_1) → P_{24}
  117. Line(G_1, O_1) → l_1
  118. OrthogonalLine(G_1, l_1) → m_1
  119. OrthogonalLine(J_1, l_1) → q_1
  120. OrthogonalLine(O_1, l_1) → r_1
  121. OrthogonalLine(R_1, l_1) → s_1
  122. OrthogonalLine(E_1, m_1) → t_1
  123. OrthogonalLine(F_1, m_1) → a_2
  124. Point(a_1) → C
  125. OrthogonalLine(F_2, d) → i_1
  126. OrthogonalLine(C, d) → j_1
  127. Intersect(d, i_1) → D_2
  128. Intersect(d, j_1) → E_2
  129. Min(180, ((a - 1) * 180)) → r2
  130. Min(270, ((a - 1) * 270)) → r3
  131. Rotate(T, r1°, T) → T'_4
  132. Rotate(F_2, r1°, T) → G_2
  133. Rotate(D_2, r1°, T) → H_2
  134. Rotate(T, r2°, T) → T'_5
  135. Rotate(F_2, r2°, T) → J_2
  136. Rotate(D_2, r2°, T) → K_2
  137. Rotate(T, r3°, T) → T'_6
  138. Rotate(F_2, r3°, T) → N_2
  139. Rotate(D_2, r3°, T) → O_2
  140. Rotate(F_2, r1°, T) → Q_2
  141. Rotate(C, r1°, T) → C'
  142. Rotate(E_2, r1°, T) → U_2
  143. Rotate(D_2, r1°, T) → V_2
  144. Rotate(F_2, r2°, T) → Z_2
  145. Rotate(C, r2°, T) → C'_1
  146. Rotate(E_2, r2°, T) → D_3
  147. Rotate(D_2, r2°, T) → F_3
  148. Rotate(F_2, r3°, T) → H_3
  149. Rotate(C, r3°, T) → C'_2
  150. Rotate(E_2, r3°, T) → M_3
  151. Rotate(D_2, r3°, T) → O_3
  152. Rotate(C, r1°, T) → C'_3
  153. Rotate(L, r1°, T) → L'_1
  154. Rotate(K_1, r1°, T) → Q_3
  155. Rotate(E_2, r1°, T) → V_3
  156. Rotate(C, r2°, T) → C'_4
  157. Rotate(L, r2°, T) → L'_3
  158. Rotate(K_1, r2°, T) → Z_3
  159. Rotate(E_2, r2°, T) → C_4
  160. Rotate(C, r3°, T) → C'_5
  161. Rotate(L, r3°, T) → L'_4
  162. Rotate(K_1, r3°, T) → H_4
  163. Rotate(E_2, r3°, T) → J_4
  164. Mirror(C'_4, a_1) → F_5
  165. Mirror(L'_3, a_1) → G_5
  166. Mirror(Z_3, a_1) → J_5
  167. Mirror(C_4, a_1) → N_5
  168. Mirror(Z_2, a_1) → O_5
  169. Mirror(C'_1, a_1) → P_5
  170. Mirror(D_3, a_1) → T_5
  171. Mirror(F_3, a_1) → U_5
  172. Mirror(T'_5, a_1) → W_5
  173. Mirror(J_2, a_1) → Z_5
  174. Mirror(K_2, a_1) → A_6
  175. Mirror(C, a_1) → C'_6
  176. Mirror(L, a_1) → L'_5
  177. Mirror(K_1, a_1) → R_6
  178. Mirror(E_2, a_1) → S_6
  179. Mirror(F_2, a_1) → T_6
  180. Mirror(C, a_1) → C'_7
  181. Mirror(E_2, a_1) → U_6
  182. Mirror(D_2, a_1) → V_6
  183. Mirror(T, a_1) → T'_{10}
  184. Mirror(F_2, a_1) → W_6
  185. Mirror(D_2, a_1) → Z_6
  186. Min(180, ((a - 5.5) * 180)) → r4
  187. Intersect(t, i) → H_{15}
  188. Midpoint(H_{15}, D) → J_{15}
  189. Vector(S, D) → u
  190. Vector(S, L) → w
  191. Line(S, O) → q
  192. Line(F, M) → r
  193. Line(D_1, V) → s
  194. Mirror(pic1, s) → pic1'
  195. Mirror(pic1', r) → pic1''
  196. Mirror(pic1'', q) → pic1'''
  197. Mirror(pic1''', e) → pic2
  198. Mirror(pic2, t) → pic2'
  199. Mirror(pic1''', t) → pic3
  200. Mirror(pic1'', t) → pic1'''_1
  201. Mirror(pic1', t) → pic1''_1
  202. Mirror(pic1, t) → pic1'_1
  203. Mirror(pic1'_1, a_1) → pic4
  204. Mirror(pic2', a_1) → pic2''
  205. Mirror(pic2'', b_1) → pic2'''
  206. Mirror(pic2''', h) → pic5
  207. Mirror(pic2'', e) → pic2'''_1
  208. Mirror(pic2''', e) → pic6
  209. Mirror(pic5, e) → pic5'
  210. Mirror(pic5', q) → pic5''
  211. Mirror(pic6, q) → pic6'
  212. Mirror(pic2'''_1, q) → pic7
  213. Mirror(pic7, r) → pic7'
  214. Mirror(pic6', r) → pic6''
  215. Mirror(pic4, b_1) → pic4'
  216. Mirror(pic5'', r) → pic5'''
  217. Mirror(pic5''', s) → pic8
  218. Intersect(t, m_1) → B_2
  219. Intersect(a_1, m_1) → P_2
  220. Intersect(b_1, m_1) → R_2
  221. Intersect(h, m_1) → S_2
  222. Intersect(a_2, q) → T_2
  223. Intersect(t_1, q) → W_2
  224. Intersect(l_1, q) → A_3
  225. Intersect(t, q_1) → B_3
  226. Intersect(q_1, a_1) → C_3
  227. Intersect(b_1, q_1) → E_3
  228. Intersect(h, q_1) → G_3
  229. Intersect(a_2, r) → I_3
  230. Intersect(r, t_1) → J_3
  231. Intersect(r, l_1) → K_3
  232. Intersect(r_1, a_1) → L_3
  233. Intersect(b_1, r_1) → N_3
  234. Intersect(h, r_1) → P_3
  235. Intersect(t, r_1) → R_3
  236. Intersect(l_1, s) → S_3
  237. Intersect(s, t_1) → T_3
  238. Intersect(s, a_2) → U_3
  239. Vector(U, P) → z
  240. Vector(U, O) → u_1

AI Drawing Prompt Reference

On the same plane, construct two related lines: draw line b through point A, then draw the related line e through point G. Choose a point J on line e. From J, drop a perpendicular to line e to obtain a point, and also draw perpendiculars from the corresponding points relative to the same reference line e and through G to create additional points. Use the intersection of two perpendicular lines to get a key point E, then repeatedly form midpoints between E and several constructed points (such as H, I, G, and J), creating a multi-level midpoint chain. Finally, mirror the last constructed midpoint points across the line h to obtain their symmetric counterparts.
The prompt above was compiled from the original drawing commands and is for reference only.