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Optimization and Special Points

Optimization and Special Points
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2020-06-23 17:04
TransformationOptimizationGeometric Construction
This example demonstrates optimization problems and special points. By constructing specific geometric shapes, it shows how to solve practical problems using commands like rotation, intersection, segment, polygon, and transformation. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Rotate(A + (1 / sin(67.5°) * (B - A)), 22.5°, A) → BARB
  2. Rotate(B, 135°, A) → BBA
  3. Rotate(B, 45°, A) → ABR
  4. Rotate(B, 45°, A) → ABSM
  5. Rotate(A + (sin(45°) / sin(112.5°) * (B - A)), 22.5°, A) → ABT
  6. Rotate(A, (-(90°)), B) → AB
  7. Rotate(B, (-(135°)), A) → ABA
  8. Rotate(A + (sin(45°) / sin(22.5°) * (B - A)), 112.5°, A) → ABAB
  9. Intersect(Line[A, Rotate[B, 45°, A]], Line[B, Rotate[A, (-(45°)), B]]) → ABH
  10. Intersect(Line[A, Rotate[B, 67.5°, A]], Line[B, Rotate[A, (-(67.5°)), B]]) → ABHR
  11. Rotate(B, 90°, A) → BA
  12. Rotate(B, 135°, A) → BAR
  13. Intersect(OrthogonalLine[B, Line[A, B]], OrthogonalLine[BBA, Line[BBA, A]]) → BBAB
  14. Segment(A, ABT) → t9pijl1
  15. Segment(A, B) → t9pijl2
  16. Segment(A, ABR) → t9pijl3
  17. Segment(A, BBAB) → t3pijl1
  18. Segment(B, BBAB) → t3pijl2
  19. Segment(BBA, BBAB) → t3pijl3
  20. Segment(BBA, A) → t3vet1
  21. Segment(A, B) → t3vet2
  22. Intersect(Line[C, Rotate[DB, 45°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[DB, (-(45°)), Line[E, E + (0, 0, 1)]]]) → CESB
  23. Intersect(Line[C, Rotate[DB, 22.5°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[DB, (-(22.5°)), Line[E, E + (0, 0, 1)]]]) → CELB
  24. Intersect(Line[C, Rotate[DB, 90°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[DB, (-(90°)), Line[E, E + (0, 0, 1)]]]) → CEA
  25. Polygon(A, B, ABT, ABR) → SmBiped9, smb92, smb91, smb94, smb93
  26. Dilate(A, 0.75, Centroid[SmBiped9]) → A'
  27. Dilate(B, 0.75, Centroid[SmBiped9]) → B'
  28. Dilate(ABT, 0.75, Centroid[SmBiped9]) → ABT'
  29. Dilate(ABR, 0.75, Centroid[SmBiped9]) → ABR'
  30. Polygon(A, B, BARB, BBA) → Almond3, alm31, alm32, alm33, alm34
  31. Polygon(A', B', ABT', ABR') → SmBiped9', a', b', abt', abr'
  32. Dilate(A, 0.75, Centroid[Almond3]) → A'_1
  33. Dilate(B, 0.75, Centroid[Almond3]) → B'_1
  34. Dilate(BBAB, 0.75, Centroid[Almond3]) → BBAB'
  35. Dilate(BBA, 0.75, Centroid[Almond3]) → BBA'
  36. Polygon(A'_1, B'_1, BBAB', BBA') → Almond3', a'_1, b'_1, bbab', bba'
  37. Polygon(D, E, EE, DD) → achter1, d, e, ee, dd
  38. Polygon(C, CC, DD, D) → achter2, c, cc, f, d_1
  39. Polygon(EE, ECA, DD) → EAlmond3boven1, g, i, eca
  40. Polygon(CC, ECA, DD) → EAlmond3boven2, j, k, l
  41. Rotate(A + (sin(22.5°) / sin(112.5°) * (B - A)), 45°, A) → ABL
  42. Polygon(A, B, AB, BA) → Square1, sq11, sq13, sq12, sq14
  43. Dilate(A, 0.75, Centroid[Square1]) → A'_2
  44. Dilate(B, 0.75, Centroid[Square1]) → B'_2
  45. Dilate(AB, 0.75, Centroid[Square1]) → AB'
  46. Dilate(BA, 0.75, Centroid[Square1]) → BA'
  47. Polygon(A'_2, B'_2, AB', BA') → Square1', a'_2, b'_2, ab', ba'
  48. Rotate(A, (-(45°)), B) → ABRH
  49. Polygon(A, B, ABRH, BAR) → Rhombus2, rh21, rh23, rh22, rh24
  50. Dilate(A, 0.75, Centroid[Rhombus2]) → A'_3
  51. Dilate(B, 0.75, Centroid[Rhombus2]) → B'_3
  52. Dilate(ABRH, 0.75, Centroid[Rhombus2]) → ABRH'
  53. Dilate(BAR, 0.75, Centroid[Rhombus2]) → BAR'
  54. Polygon(A'_3, B'_3, ABRH', BAR') → Rhombus2', a'_3, b'_3, abrh', bar'
  55. Rotate(A + (sin(67.5°) / sin(45°) * (B - A)), 67.5°, A) → BAJ
  56. Polygon(A, B, BAJ, BBA) → Jug4, jug41, jug42, jug43, jug44
  57. Dilate(A, 0.75, Centroid[Jug4]) → A'_4
  58. Dilate(B, 0.75, Centroid[Jug4]) → B'_4
  59. Dilate(BAJ, 0.75, Centroid[Jug4]) → BAJ'
  60. Dilate(BBA, 0.75, Centroid[Jug4]) → BBA'_1
  61. Polygon(A'_4, B'_4, BAJ', BBA'_1) → Jug4', a'_4, b'_4, baj', bba'_1
  62. Polygon(A, B, ABH) → HSquare5, hsq52, hsq51, hsq53
  63. Dilate(A, 0.75, Centroid[HSquare5]) → A'_5
  64. Dilate(B, 0.75, Centroid[HSquare5]) → B'_5
  65. Dilate(ABH, 0.75, Centroid[HSquare5]) → ABH'
  66. Polygon(A'_5, B'_5, ABH') → HSquare5', abh', a'_5, b'_5
  67. Polygon(A, B, ABHR) → HRhombus6, hrh62, hrh61, hrh63
  68. Dilate(A, 0.75, Centroid[HRhombus6]) → A'_6
  69. Dilate(B, 0.75, Centroid[HRhombus6]) → B'_6
  70. Dilate(ABHR, 0.75, Centroid[HRhombus6]) → ABHR'
  71. Polygon(A'_6, B'_6, ABHR') → HRhombus6', abhr', a'_6, b'_6
  72. Polygon(A, B, ABL, BA) → LBiped10, lb101, lb103, lb102, lb104
  73. Dilate(A, 0.75, Centroid[LBiped10]) → A'_7
  74. Dilate(B, 0.75, Centroid[LBiped10]) → B'_7
  75. Dilate(ABL, 0.75, Centroid[LBiped10]) → ABL'
  76. Dilate(BA, 0.75, Centroid[LBiped10]) → BA'_1
  77. Polygon(A'_7, B'_7, ABL', BA'_1) → LBiped10', a'_7, b'_7, abl', ba'_1
  78. Polygon(A, B, BA) → HSquare11, hsq113, hsq111, hsq112
  79. Dilate(A, 0.75, Centroid[HSquare11]) → A'_8
  80. Dilate(B, 0.75, Centroid[HSquare11]) → B'_8
  81. Dilate(BA, 0.75, Centroid[HSquare11]) → BA'_2
  82. Polygon(A'_8, B'_8, BA'_2) → HSquare11', ba'_2, a'_8, b'_8
  83. Polygon(A, B, ABR) → HRhombus12, hrh121, hrh123, hrh122
  84. Dilate(A, 0.75, Centroid[HRhombus12]) → A'_9
  85. Dilate(B, 0.75, Centroid[HRhombus12]) → B'_9
  86. Dilate(ABR, 0.75, Centroid[HRhombus12]) → ABR'_1
  87. Polygon(A'_9, B'_9, ABR'_1) → HRhombus12', abr'_1, a'_9, b'_9
  88. Polygon(ABA, A, B, ABAB) → SmBiped13, smb132, smb131, smb134, smb133
  89. Dilate(ABA, 0.75, Centroid[SmBiped13]) → ABA'
  90. Dilate(A, 0.75, Centroid[SmBiped13]) → A'_{10}
  91. Dilate(B, 0.75, Centroid[SmBiped13]) → B'_{10}
  92. Dilate(ABAB, 0.75, Centroid[SmBiped13]) → ABAB'
  93. Polygon(ABA', A'_{10}, B'_{10}, ABAB') → SmBiped13', aba', a'_{10}, b'_{10}, abab'
  94. Polygon(A, B, BARB, ABR) → Almond14, m, n, p, q
  95. Dilate(A, 0.75, Centroid[Almond14]) → A'_{11}
  96. Dilate(B, 0.75, Centroid[Almond14]) → B'_{11}
  97. Dilate(BARB, 0.75, Centroid[Almond14]) → BARB'
  98. Dilate(BAR, 0.75, Centroid[Almond14]) → BAR'_1
  99. Segment(A, AB) → sq1pijl1
  100. Segment(B, AB) → sq1pijl2
  101. Segment(A, B) → sqvet1
  102. Segment(A, BA) → sqvet2
  103. Segment(BA, AB) → sq1pijl3
  104. Segment(A, B) → sq7pijl3
  105. Segment(A, BA) → sq7pijl2
  106. Segment(A, ABRH) → rh2pijl1
  107. Segment(B, ABRH) → rh2pijl2
  108. Segment(BAR, ABRH) → rh2pijl3
  109. Segment(A, B) → rh2vet1
  110. Segment(A, BAR) → rh2vet2
  111. Segment(A, B) → rh8pijl2
  112. Segment(A, BAR) → rh8pijl3
  113. Segment(A, BARB) → alm3pijl1
  114. Segment(B, BARB) → alm3pijl2
  115. Segment(BBA, BARB) → alm3pijl3
  116. Segment(A, B) → alm3vet1
  117. Segment(A, BBA) → alm3vet2
  118. Segment(A, BAJ) → jug4pijl1
  119. Segment(B, BAJ) → jug4pijl2
  120. Segment(BBA, BAJ) → jug4pijl3
  121. Segment(A, B) → jug4vet1
  122. Segment(A, BBA) → jug4vet2
  123. Segment(Midpoint[A, B], ABH) → hsq5pijl1
  124. Segment(A, ABH) → hsq5pijl2
  125. Segment(B, ABH) → hsq5pijl3
  126. Segment(A, B) → hsq5vet1
  127. Segment(Midpoint[A, B], ABHR) → hrh6pijl1
  128. Segment(A, ABHR) → hrh6pijl2
  129. Segment(B, ABHR) → hrh6pijl3
  130. Segment(A, B) → hrh6vet1
  131. Segment(A, ABT) → smb9pijl1
  132. Segment(A, B) → smb9pijl2
  133. Segment(A, ABR) → smb9pijl3
  134. Segment(A, ABL) → lb10pijl1
  135. Segment(A, B) → lb10pijl2
  136. Segment(A, BA) → lb10pijl3
  137. Segment(A, Midpoint[BA, B]) → hsq11pijl1
  138. Segment(A, B) → hsq11pijl2
  139. Segment(A, BA) → hsq11pijl3
  140. Segment(A, Midpoint[ABR, B]) → hrh12pijl1
  141. Segment(A, ABR) → hrh12pijl2
  142. Segment(A, B) → hrh12pijl3
  143. Segment(A, ABAB) → smb13pijl1
  144. Segment(B, ABAB) → smb13pijl2
  145. Segment(ABA, ABAB) → smb13pijl3
  146. Segment(A, B) → smb13vet1
  147. Segment(A, ABA) → smb13vet2
  148. Segment(A, BARB) → alm14pijl1
  149. Polygon(EE, EC, DD) → ESquare1boven1, i_1, j_1, ec
  150. Polygon(CC, EC, DD) → ESquare1boven2, k_1, l_1, m_1
  151. Intersect(Line[C, Line[DB, E]], Line[E, Line[DB, C]]) → CE
  152. Polygon(DO, E, CE) → ESquare7boven1, r, s, t
  153. Polygon(DO, CE, C) → ESquare7boven2, a, b, f_1
  154. Polygon(EE, ECJ, DD) → EJug4boven1, q_1, r_1, ecj
  155. Polygon(CC, ECJ, DD) → EJug4boven2, s_1, t_1, f_2
  156. Polygon(CC, DD, CDHS) → EHSquare5boven, cdhs, g_2, h_2
  157. Polygon(CC, DD, CDHR) → EHRhombus6boven, cdhr, i_2, j_2
  158. Polygon(DO, E, CESB) → ESmBiped9boven1, g_1, h_1, n_1
  159. Polygon(DO, CESB, C) → ESmBiped9boven2, p_1, a_1, b_1
  160. Polygon(DO, E, CELB) → ELBiped10boven1, c_1, e_1, k_2
  161. Polygon(DO, CELB, C) → ELBiped10boven2, l_2, m_2, n_2
  162. Polygon(C, DO, E) → EHSquare11boven11, p_2, q_2, r_2
  163. Polygon(EE, ECSB, DD) → ESmbiped13boven1, s_2, t_2, ecsb
  164. Polygon(DD, ECSB, CC) → ESmbiped13boven2, d_2, f_3, g_3
  165. Polygon(DO, E, CEA) → EAlmond14onder1, a_2, b_2, c_2
  166. Polygon(DO, CEA, C) → EAlmond14onder2, e_2, h_3, i_3
  167. Dilate(A, 0.75, Centroid[Almond14]) → A'_{12}
  168. Dilate(B, 0.75, Centroid[Almond14]) → B'_{12}
  169. Dilate(BARB, 0.75, Centroid[Almond14]) → BARB'_1
  170. Dilate(ABR, 0.75, Centroid[Almond14]) → ABR'_2
  171. Polygon(A'_{12}, B'_{12}, BARB'_1, ABR'_2) → Almond14', a'_{12}, b'_{12}, barb'_1, abr'_2
  172. Segment(A, B) → alm14pijl2
  173. Segment(A, ABR) → alm14pijl3
  174. Intersect(xAxis, yAxis) → F
  175. RightSqrt2(F) → G
  176. LeftSqrt2(F) → H
  177. Right1(G) → I
  178. Left1(H) → J
  179. TSquare1(J, H) → K, L, vierhoek1, vierhoek2, j_3, k_3, l_3, m_3, n_3
  180. HSquare5(H, F) → M, driehoek1, driehoek2, p_3, q_3, r_3, s_3
  181. HSquare5(F, G) → N, driehoek3, driehoek4, t_3, a_3, b_3, c_3
  182. Up1(I) → O
  183. TSquare1(I, O) → P, Q, vierhoek3, vierhoek4, d_3, e_3, f_4, g_4, h_4
  184. Left1(L) → R
  185. Right1(O) → S
  186. Up1(S) → T
  187. TSquare1(R, L) → U, V, vierhoek5, vierhoek6, i_4, j_4, k_4, l_4, m_4
  188. TSquare1(S, T) → W, Z, vierhoek7, vierhoek8, n_4, p_4, q_4, r_4, s_4
  189. UpSqrt2(V) → A_1
  190. UpSqrt2(T) → B_1
  191. HSquare5(A_1, V) → C_1, driehoek5, driehoek6, t_4, a_4, b_4, c_4
  192. HSquare5(T, B_1) → D_1, driehoek7, driehoek8, d_4, e_4, f_5, g_5
  193. TSmBiped9(A_1, C_1) → E_1, F_1, vierhoek9, vierhoek10, h_5, i_5, j_5
  194. TSmBiped9(V, U) → G_1, H_1, vierhoek11, vierhoek12, k_5, l_5, m_5
  195. TSmBiped9(L, K) → I_1, J_1, vierhoek13, vierhoek14, n_5, p_5, q_5
  196. TSmBiped9(L, I_1) → K_1, L_1, vierhoek15, vierhoek16, r_5, s_5, t_5
  197. TSmBiped9(H, M) → M_1, N_1, vierhoek17, vierhoek18, a_5, b_5, c_5
  198. TSmBiped9(F, N) → O_1, P_1, vierhoek19, vierhoek20, d_5, e_5, f_6
  199. TSmBiped9(F, O_1) → Q_1, R_1, vierhoek21, vierhoek22, g_6, h_6, i_6
  200. TSmBiped9(G, P) → S_1, T_1, vierhoek23, vierhoek24, j_6, k_6, l_6
  201. TSmBiped9(O, W) → U_1, V_1, vierhoek25, vierhoek26, m_6, n_6, p_6
  202. TSmBiped9(O, U_1) → W_1, Z_1, vierhoek27, vierhoek28, q_6, r_6, s_6
  203. TSmBiped9(T, D_1) → A_2, B_2, vierhoek29, vierhoek30, t_6, a_6, b_6
  204. Left1(B_1) → C_2
  205. TSmBiped9(B_1, C_2) → D_2, E_2, vierhoek31, vierhoek32, c_6, d_6, e_6
  206. TAlmond3(F_1, C_1) → F_2, G_2, vierhoek33, vierhoek34, f_7, g_7, h_7, i_7, j_7
  207. TAlmond3(H_1, U) → H_2, I_2, vierhoek35, vierhoek36, k_7, l_7, m_7, n_7, p_7
  208. TAlmond3(L_1, I_1) → J_2, K_2, vierhoek37, vierhoek38, q_7, r_7, s_7, t_7, a_7
  209. TAlmond3(J_1, K) → L_2, M_2, vierhoek39, vierhoek40, b_7, c_7, d_7, e_7, f_8
  210. TAlmond3(N_1, M) → N_2, O_2, vierhoek41, vierhoek42, g_8, h_8, i_8, j_8, k_8
  211. TAlmond3(R_1, O_1) → P_2, Q_2, vierhoek43, vierhoek44, l_8, m_8, n_8, p_8, q_8
  212. TAlmond3(P_1, N) → R_2, S_2, vierhoek45, vierhoek46, r_8, s_8, t_8, a_8, b_8
  213. TAlmond3(T_1, P) → T_2, U_2, vierhoek47, vierhoek48, c_8, d_8, e_8, f_9, g_9
  214. TAlmond3(Z_1, U_1) → V_2, W_2, vierhoek49, vierhoek50, h_9, i_9, j_9, k_9, l_9
  215. TAlmond3(V_1, W) → Z_2, A_3, vierhoek51, vierhoek52, m_9, n_9, p_9, q_9, r_9
  216. TAlmond3(B_2, D_1) → B_3, C_3, vierhoek53, vierhoek54, s_9, t_9, a_9, b_9, c_9
  217. TAlmond3(E_2, C_2) → D_3, E_3, vierhoek55, vierhoek56, d_9, e_9, f_{10}, g_{10}, h_{10}
  218. TLBiped10(I_1, L_2) → F_3, G_3, vierhoek57, vierhoek58, i_{10}, j_{10}, k_{10}
  219. TLBiped10(O_1, R_2) → H_3, I_3, vierhoek59, vierhoek60, l_{10}, m_{10}, n_{10}
  220. TLBiped10(U_1, Z_2) → J_3, K_3, vierhoek61, vierhoek62, p_{10}, q_{10}, r_{10}
  221. EHSquare5(A_1, V) → L_3, M_3, N_3, vierhoek63, driehoek9
  222. ESquare1(V, R, L) → O_3, P_3, Q_3, vierhoek64, vierhoek65, driehoek10, driehoek11
  223. ESquare1(L, J, H) → R_3, S_3, T_3, vierhoek66, vierhoek67, driehoek12, driehoek13
  224. EHSquare5(H, F) → U_3, V_3, W_3, vierhoek68, driehoek14
  225. EHSquare5(F, Q) → Z_3, A_4, B_4, vierhoek69, driehoek15
  226. ESquare1(Q, I, Z) → C_4, D_4, E_4, vierhoek70, vierhoek71, driehoek16, driehoek17
  227. ESquare1(Z, S, T) → F_4, G_4, H_4, vierhoek72, vierhoek73, driehoek18, driehoek19
  228. EHSquare5(T, B_1) → I_4, J_4, K_4, vierhoek74, driehoek20
  229. ESmBiped9(L_4, L_3, N_3) → M_4, driehoek21, driehoek22
  230. ESmBiped9(N_3, O_3, Q_3) → N_4, driehoek23, driehoek24
  231. ESmBiped9(Q_3, R_3, O_4) → P_4, driehoek25, driehoek26
  232. ESmBiped9(O_4, R_3, T_3) → Q_4, driehoek27, driehoek28
  233. ESmBiped9(T_3, U_3, W_3) → R_4, driehoek29, driehoek30
  234. ESmBiped9(W_3, Z_3, S_4) → T_4, driehoek31, driehoek32
  235. ESmBiped9(S_4, Z_3, B_4) → U_4, driehoek33, driehoek34
  236. ESmBiped9(B_4, C_4, E_4) → V_4, driehoek35, driehoek36
  237. ESmBiped9(E_4, F_4, W_4) → Z_4, driehoek37, driehoek38
  238. ESmBiped9(W_4, F_4, H_4) → A_5, driehoek39, driehoek40
  239. ESmBiped9(H_4, I_4, K_4) → B_5, driehoek41, driehoek42
  240. ESmBiped9(K_4, J_4, C_5) → D_5, driehoek43, driehoek44
  241. EAlmond3(L_4, M_4, N_3) → E_5, F_5, G_5, vierhoek75, vierhoek76, driehoek45, driehoek46
  242. EAlmond3(N_3, N_4, Q_3) → H_5, I_5, J_5, vierhoek77, vierhoek78, driehoek47, driehoek48
  243. EAlmond3(Q_3, P_4, O_4) → K_5, L_5, M_5, vierhoek79, vierhoek80, driehoek49, driehoek50
  244. EAlmond3(O_4, Q_4, T_3) → N_5, O_5, P_5, vierhoek81, vierhoek82, driehoek51, driehoek52
  245. EAlmond3(T_3, R_4, W_3) → Q_5, R_5, S_5, vierhoek83, vierhoek84, driehoek53, driehoek54
  246. EAlmond3(W_3, T_4, S_4) → T_5, U_5, V_5, vierhoek85, vierhoek86, driehoek55, driehoek56
  247. EAlmond3(S_4, U_4, B_4) → W_5, Z_5, A_6, vierhoek87, vierhoek88, driehoek57, driehoek58
  248. EAlmond3(B_4, V_4, E_4) → B_6, C_6, D_6, vierhoek89, vierhoek90, driehoek59, driehoek60
  249. EAlmond3(E_4, Z_4, W_4) → E_6, F_6, G_6, vierhoek91, vierhoek92, driehoek61, driehoek62
  250. EAlmond3(W_4, A_5, H_4) → H_6, I_6, J_6, vierhoek93, vierhoek94, driehoek63, driehoek64
  251. EAlmond3(H_4, B_5, K_4) → K_6, L_6, M_6, vierhoek95, vierhoek96, driehoek65, driehoek66
  252. EAlmond3(K_4, D_5, C_5) → N_6, O_6, P_6, vierhoek97, vierhoek98, driehoek67, driehoek68
  253. ELBiped10(M_5, N_5, V_5) → Q_6, driehoek69, driehoek70
  254. ELBiped10(V_5, W_5, G_6) → R_6, driehoek71, driehoek72
  255. ELBiped10(G_6, H_6, P_6) → S_6, driehoek73, driehoek74