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优化问题与特殊点

优化问题与特殊点
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2020-06-23 17:04
变换最值优化几何作图
该范例演示了优化问题和特殊点的概念。通过构造特定的几何图形，展示了如何利用旋转、交点、线段等命令来解决实际问题。使用了旋转、交点、线段、多边形、变换等命令。适合高中数学教学和学习。

GeoGebra 命令及参数

  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