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Trefoil Arch Construction

Trefoil Arch Construction
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2023-03-17 22:21
Solid GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric construction and properties of a Trefoil Arch. It involves core mathematical concepts such as segments, circles, rotations, reflections, and polygons. The construction utilizes commands like Midpoint, Segment, Rotate, Circle, and Polygon in GeoGebra. Suitable for high school mathematics education, showcasing applications of spatial geometry and transformations.

GeoGebra Commands & Parameters

  1. Midpoint(C, D) → F
  2. Midpoint(F, C) → R
  3. Midpoint(R, F) → V
  4. If(zz < 0, 10, zz) → j
  5. Segment(D, B) → e
  6. Line(F, e) → k
  7. Midpoint(C, D) → E
  8. Segment(A, C) → d
  9. Rotate(C, 359.91000000000014°, E) → C''
  10. CircumcircleArc(C, C'', S1) → g
  11. Segment(C, F) → r
  12. Mirror(D, k) → D'_1
  13. Segment(R, S1) → f
  14. Mirror(C, k) → C'
  15. Mirror(C'', k) → C'''
  16. Mirror(S1, k) → S1'
  17. CircumcircleArc(C', C''', S1') → g'
  18. Mirror(S1, k) → S1'_1
  19. Mirror(S1, k) → S1'_2
  20. Mirror(R, k) → R'
  21. Mirror(S1, k) → S1'_3
  22. Segment(R', S1'_3) → f'
  23. Circle(V, C) → e_2
  24. Mirror(C, k) → C'_3
  25. Mirror(S1, k) → S1'_4
  26. Mirror(V, k) → V'
  27. Midpoint(R, C) → W
  28. Mirror(C, k) → C'_4
  29. Mirror(V, k) → V'_1
  30. Circle(C, i) → h
  31. Intersect(h, r, 1) → G
  32. Mirror(G, k) → G'
  33. Circle(G, G') → p
  34. Intersect(p, k, 1) → H
  35. Polygon(G, H, G') → poly2, g'_1, g_1, h_1
  36. Rotate(C, Max[-2.09, (((-(j - 2)) * 3.14) * 2 / 9)], G) → C'_1
  37. Rotate(C'_1, (-Min[5.24, ((j - 5) * 5.24 / 3)]), H) → I
  38. Segment(I, H) → l
  39. CircleArc(H, I, C'_1) → q
  40. Segment(C'_1, G) → s
  41. CircleArc(G, C'_1, C) → t
  42. Mirror(G, k) → G'_1
  43. Mirror(C'_1, k) → J
  44. Mirror(C, k) → C'_2
  45. Arc(Mirror[Circle[G, C'_1], k], J, C'_2) → t'
  46. Mirror(C'_1, k) → K
  47. Mirror(G, k) → G'_2
  48. Segment(K, G'_2) → s'
  49. Midpoint(C, D) → L
  50. Segment(C, L) → a_1
  51. Circle(C, m) → c_1
  52. Intersect(c_1, a_1, 1) → M
  53. Line(M, k) → b_1
  54. Circle(M, C) → d_1
  55. Intersect(d_1, b_1, 1) → N
  56. OrthogonalLine(N, k) → e_1
  57. Intersect(k, e_1) → O
  58. Circle(N, O) → f_1
  59. Intersect(f_1, b_1, 1) → P
  60. Intersect(c_1, e_1, 1) → Q
  61. Rotate(C, Min[1.57, ((n - 2) * 1.57 / 2)], Q) → C'_5
  62. Rotate(N, Min[1.57, ((n - 5) * 1.57 / 3)], P) → N'
  63. CircleArc(Q, C, C'_5) → k_1
  64. Segment(Q, C'_5) → i_1
  65. CircleArc(P, N, N') → p_1
  66. Segment(P, N') → j_1
  67. Mirror(P, k) → P'
  68. Mirror(N, k) → N'_1
  69. Mirror(N', k) → N''
  70. Arc(Mirror[Circle[P, N], k], N'_1, N'') → q_1
  71. Mirror(Q, k) → Q'
  72. Mirror(C, k) → C'_6
  73. Mirror(C'_5, k) → A_1
  74. Arc(Mirror[Circle[Q, C], k], C'_6, A_1) → r_1
  75. Mirror(P, k) → P'_1
  76. Mirror(N', k) → N''_1
  77. Segment(P'_1, N''_1) → l_1
  78. Mirror(Q, k) → Q'_1
  79. Mirror(C'_5, k) → B_1
  80. Segment(Q'_1, B_1) → m_1
  81. Segment(T2, T1) → f_2
  82. Mirror(D, f_2) → D'
  83. Mirror(B, f_2) → B'
  84. Segment(D', B') → e'
  85. Mirror(A, f_2) → A'
  86. Mirror(C, f_2) → C'_7
  87. Segment(A', C'_7) → d'
  88. Vector(A, B') → u_1
  89. Translate(pic1, u_1) → pic1'
  90. Translate(pic1, u_1) → pic1'_1
  91. Midpoint(C, D) → D_1

AI Drawing Prompt Reference

This is a geometric figure composed of multiple circles and line segments, including a central point, symmetric arcs, and connecting lines. The figure is created through rotations, reflections, and arc constructions, demonstrating spatial geometry and transformations. The construction order involves determining midpoints, drawing line segments, rotating points, drawing arcs, and multiple reflections, resulting in a complex symmetric figure.
The prompt above was compiled from the original drawing commands and is for reference only.