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Geometric Construction of the Brazil Flag Logo

Geometric Construction of the Brazil Flag Logo
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2020-08-03 00:42
PolygonVectorCoordinate GeometryTransformationGeometric Construction
This example demonstrates how to construct the green pentagon star on the Brazil flag using geometric construction methods. The core math concepts involved include angles, circles, polygons, vectors, coordinate geometry, and geometric transformations. Key construction steps involve commands like LineBisector, Vector, OrthogonalVector, Intersect, Line, OrthogonalLine, Mirror, Circle, Segment, Polygon, Point, Rotate, Angle, CircleArc, Spline, Sequence, Element, and Length from GeoGebra. It is suitable for high school students learning geometric constructions, coordinate geometry, and transformations, as well as for math clubs.

GeoGebra Commands & Parameters

  1. LineBisector(A, B) → m
  2. Vector(A, B) → u
  3. OrthogonalVector(u) → v
  4. LineBisector(A_1, A) → l
  5. Intersect(m, l) → K
  6. Line(A, B) → s
  7. OrthogonalLine(D, s) → i
  8. Intersect(i, l) → J
  9. Mirror(L, m) → L'
  10. OrthogonalLine(A, s) → b
  11. OrthogonalLine(F, b) → f
  12. Mirror(f, l) → f'
  13. Intersect(m, f') → S
  14. Mirror(I, l) → I'
  15. Line(L', I') → g_1
  16. Mirror(O, l) → O'_1
  17. Circle(K, O'_1) → f_1
  18. Intersect(f_1, g_1) → F_1, G_1
  19. Mirror(G_1, m) → H_1
  20. Segment(G_1, H_1) → m_1
  21. Intersect(m, m_1) → I_1
  22. Mirror(f, l) → f'_1
  23. Intersect(i, f') → E
  24. OrthogonalLine(G, f'_1) → h
  25. OrthogonalLine(I, m) → n
  26. Polygon(L, I, L', I') → c1, l_1, i_1, l', j
  27. Point(l_1) → Z
  28. OrthogonalLine(A_1, b) → r
  29. OrthogonalLine(B, u, xOyPlane) → g
  30. Mirror(i, m) → i'
  31. Intersect(g, r) → C
  32. Mirror(F, l) → F'
  33. Polygon(A_1, C, B, A) → c4, a_1, c_3, b_2, a_2
  34. OrthogonalLine(H, f'_1) → k
  35. Intersect(i', f) → J_1
  36. Circle(K, S) → c
  37. Circle(K, M) → d
  38. Mirror(N, m) → N'
  39. Rotate(L, 15°, K) → L'_1
  40. Rotate(L, (-(10°)), K) → L'_3
  41. Angle(L'_3, K, L) → α
  42. Line(L'_3, K) → p
  43. Line(P, p) → q
  44. Mirror(q, p) → q'
  45. Intersect(d, q') → Q, R
  46. Intersect(d, q) → T, U
  47. LineBisector(Q, R) → t
  48. Polygon(Q, R, U, T) → c2, q_1, r_1, u_1, t_1
  49. Polygon(Z, I, O) → t1, o, z_1, i_2
  50. Mirror(Z, m) → Z'
  51. Mirror(I, m) → I'_1
  52. Mirror(O, m) → O'
  53. LineBisector(R, Q) → e
  54. Intersect(c, e) → B_1, C_1
  55. CircleArc(C_1, R, Q) → c_1
  56. CircleArc(C_1, U, T) → d_1
  57. Intersect(f, m) → D_1
  58. Spline({O', V, N, W, Z}, 4) → l1
  59. Mirror(l1, m) → l1'
  60. Polygon(Z, I'_1, O') → t2, o', z_2, i'_1
  61. Mirror(Z, m) → Z'_1
  62. Mirror(I'_1, m) → E_1
  63. Mirror(O', m) → O''
  64. Polygon(Z'_1, E_1, O'') → t2', o'', z'_1, a
  65. CircleArc(K, G_1, H_1) → p_1
  66. Mirror(g_1, m) → s_1
  67. Sequence(y = y(I_1) + (d1 * (y(I') - y(I_1))), d1, 0, 1, 0.04) → g_2
  68. Sequence(Segment[Intersect[Element[g_2, i], s_1], Intersect[Element[g_2, i], g_1]], i, 1, Length[g_2]) → l2
  69. Element(l2, 1) → j_1
  70. Element(l2, 2) → k_1
  71. Element(l2, 4) → j_2
  72. Element(l2, 6) → l_2
  73. Element(l2, 3) → k_2
  74. Element(l2, 7) → n_1
  75. Element(l2, 9) → b_1
  76. Element(l2, 10) → h_2
  77. Element(l2, 11) → m_2
  78. Element(l2, 12) → n_2
  79. Element(l2, 13) → p_2
  80. Element(l2, 14) → q_2
  81. Element(l2, 15) → r_2
  82. Element(l2, 16) → s_2
  83. Element(l2, 17) → t_2
  84. Element(l2, 18) → c_2
  85. Element(l2, 19) → d_2
  86. Element(l2, 20) → f_3
  87. Length(l2) → w
  88. Element(l2, 21) → g_3
  89. Element(l2, 22) → h_3
  90. Element(l2, 23) → i_3
  91. Element(l2, 24) → j_3
  92. Element(l2, 25) → k_3
  93. Element(l2, 3) → m_3
  94. Element(l2, 5) → n_3
  95. Element(l2, 8) → p_3
  96. Sequence(y = y(E_1) + (d1 * (y(O) - y(E_1))), d1, 0, 1, 0.04) → l3
  97. Sequence(Segment[Intersect[Element[l3, i], l1], Intersect[Element[l3, i], l1']], i, 1, Length[l3]) → l4
  98. Element(l4, 1) → h_1
  99. Element(l4, 2) → l_3
  100. Element(l4, 3) → q_3
  101. Element(l4, 4) → r_3
  102. Element(l4, 5) → s_3
  103. Element(l4, 6) → t_3
  104. Element(l4, 7) → b_3
  105. Element(l4, 8) → d_3
  106. Element(l4, 9) → e_3
  107. Element(l4, 10) → f_4
  108. Element(l4, 12) → g_4
  109. Element(l4, 13) → h_4
  110. Element(l4, 14) → i_4
  111. Element(l4, 15) → j_4
  112. Element(l4, 16) → k_4
  113. Element(l4, 17) → l_4
  114. Element(l4, 18) → m_4
  115. Element(l4, 19) → n_4
  116. Element(l4, 20) → p_4
  117. Element(l4, 21) → q_4
  118. Element(l4, 22) → r_4
  119. Element(l4, 23) → s_4
  120. Element(l4, 24) → t_4
  121. Element(l4, 25) → a_4

AI Drawing Prompt Reference

Construct a geometric diagram of the Brazil flag logo featuring a green five-pointed star. Start with a base segment and its perpendicular bisector, then add vectors and orthogonal lines. Use intersections, reflections, and rotations to define star vertices. Include circles centered at key points, polygons (quadrilaterals and pentagons), and multiple line segments. The final figure shows symmetric geometric relationships with angle markings and auxiliary construction lines.
The prompt above was compiled from the original drawing commands and is for reference only.