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Geometric Construction Guided Optimization Example

Geometric Construction Guided Optimization Example
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2020-09-15 11:43
CirclePolygonAngleOptimizationGeometric Construction
This example demonstrates an optimization workflow driven by geometric construction: within a configuration constrained by lines, circles, and a polygon, a target quantity defined from geometric measurements (e.g., distances or angles) changes as construction parameters vary, and an optimal configuration can be identified visually. The key mathematical ideas include optimization in a constrained geometric setting, distance and angle measurement, and generating equivalent or symmetric configurations using construction tools. Construction-wise, it is built from GeoGebra commands such as Line and OrthogonalLine to create guiding lines, Point for defining key vertices, Segment and Polygon to form the polygonal region, Circle together with Intersect to locate intersection points, Distance to compute the measurable quantities, Rotate to transform the figure, and Angle to extract the angular relations that participate in the optimization. Suitable for secondary school to early university students, and useful for teaching how geometric models connect with optimization-style reasoning.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. Point(Circle[A, 24]) → C
  4. Segment(A, C) → h
  5. Point(Circle[A, 16]) → D
  6. Segment(A, D) → i
  7. Circle(A, D) → c
  8. Intersect(c, g, 2) → E
  9. OrthogonalLine(E, g) → j
  10. OrthogonalLine(C, f) → k
  11. Segment(E, A) → l
  12. Segment(A, C) → m
  13. Intersect(j, k) → F
  14. Segment(C, F) → n
  15. Segment(F, E) → p
  16. Circle(A, Cutout) → d
  17. Circle(E, Cutout) → e
  18. Circle(F, Cutout) → q
  19. Circle(C, Cutout) → r
  20. Intersect(e, l, 1) → G
  21. Intersect(q, n, 1) → H
  22. Segment(G, H) → s
  23. Intersect(r, n, 1) → I
  24. Intersect(d, l, 1) → J
  25. Segment(I, J) → t
  26. Intersect(d, h, 1) → K
  27. Intersect(e, p, 1) → L
  28. Segment(K, L) → a
  29. Intersect(q, p, 1) → M
  30. Intersect(r, h, 1) → N
  31. Segment(M, N) → b
  32. Intersect(s, a) → O
  33. Intersect(s, b) → P
  34. Intersect(t, b) → Q
  35. Intersect(t, a) → R
  36. Segment(G, O) → f_1
  37. Segment(O, L) → g_1
  38. Segment(P, H) → i_1
  39. Segment(P, M) → j_1
  40. Segment(Q, I) → l_1
  41. Segment(R, J) → p_1
  42. Segment(R, K) → q_1
  43. Segment(Q, N) → r_1
  44. Polygon(G, O, L, E) → q1, g_2, o, l_2, e_1
  45. Polygon(J, R, K, A) → q2, j_2, r_2, k_1, a_1
  46. Polygon(N, Q, I, C) → q3, n_1, q_2, i_2, c_1
  47. Polygon(M, F, H, P) → q4, m_1, f_2, h_1, p_2
  48. Polygon(G, O, R, J, L, M, P, H, I, Q, N, K) → poly1, g_3, k_2, r_3, j_3, o_1, l_3, m_2, p_3, h_2, i_3, q_3, n_2
  49. Distance(J, G) → distanceJG
  50. Distance(L, M) → distanceLM
  51. Segment(G, J) → s_1
  52. Segment(L, M) → t_1
  53. Segment(G, E) → b_1
  54. Segment(J, A) → d_1
  55. Segment(G, J) → s_2
  56. Segment(L, M) → t_2
  57. Line(T, U) → a_2
  58. Rotate(T, -120°, U) → T'
  59. Angle(T', U, T) → α
  60. Line(U, T') → b_2
  61. Circle(U, j_3) → c_2
  62. Circle(U, l_3) → d_2
  63. Intersect(d_2, b_2, 2) → V
  64. Intersect(c_2, b_2, 2) → W
  65. Line(W, a_2) → e_2
  66. Intersect(d_2, a_2, 1) → Z
  67. Line(Z, b_2) → f_3
  68. Circle(U, g_1) → c_3
  69. OrthogonalLine(U, a_2) → h_3
  70. Line(W, h_3) → k_3
  71. Line(Z, h_3) → m_3
  72. Intersect(e_2, f_3) → A_1
  73. Line(A_1, h_3) → n_3
  74. Intersect(c_3, h_3, 2) → B_1
  75. Line(B_1, b_2) → s_3
  76. Line(B_1, a_2) → t_3
  77. Intersect(m_3, t_3) → C_1
  78. Line(C_1, s_3) → a_3
  79. Intersect(k_3, s_3) → D_1
  80. Line(D_1, t_3) → b_3
  81. Intersect(n_3, a_3) → E_1
  82. Intersect(f_3, t_3) → F_1
  83. Intersect(e_2, s_3) → G_1
  84. Segment(Z, U) → d_3
  85. Segment(U, B_1) → e_3
  86. Segment(U, W) → f_4
  87. Segment(W, D_1) → g_4
  88. Segment(D_1, G_1) → h_4
  89. Segment(G_1, B_1) → i_4
  90. Segment(B_1, F_1) → j_4
  91. Segment(F_1, C_1) → k_4
  92. Segment(C_1, Z) → l_4
  93. Segment(C_1, E_1) → m_4
  94. Segment(E_1, A_1) → n_4
  95. Segment(A_1, F_1) → p_4
  96. Segment(E_1, D_1) → q_4
  97. Segment(G_1, A_1) → r_4
  98. Polygon(F_1, C_1, E_1, A_1) → q5, s_4, t_4, a_4, b_4
  99. Polygon(E_1, D_1, G_1, A_1) → q6, c_4, d_4, e_4, f_5
  100. Polygon(A_1, G_1, B_1, F_1) → q7, g_5, h_5, i_5, j_5
  101. Polygon(B_1, D_1, W, U) → q8, k_5, l_5, w, u
  102. Polygon(C_1, Z, U, B_1) → q9, m_5, z_1, u_1, n_5

AI Drawing Prompt Reference

Draw a geometric figure composed of lines, circles, and polygons. Start by drawing two parallel lines f and g. Then, take points C and D on circles, and connect AC and AD to form segments. Next, draw circle c and find its intersection E with g. Use rotations and perpendicular lines to construct a series of intersection points F, G, H, I, J, K, L, M, and N. Finally, connect these points to form several polygons, including quadrilateral GOLE, quadrilateral JRAK, quadrilateral NQIC, and quadrilateral MFHP.
The prompt above was compiled from the original drawing commands and is for reference only.