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Schwarz minimal triangle

Schwarz minimal triangle
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2015-11-30 00:51
TriangleOptimizationGeometric Construction
This example demonstrates the construction and properties of the Schwarz minimal triangle, the triangle with the minimum area for a given perimeter. It involves core mathematical concepts such as perimeter, area, trigonometry, coordinate geometry, and geometric construction. The construction uses GeoGebra commands/objects like Polygon, OrthogonalLine, Intersect, Mirror, and Segment. Suitable for high school and above students to understand geometric optimization problems and construction methods.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. OrthogonalLine(C, c) → d
  3. OrthogonalLine(B, b) → e
  4. OrthogonalLine(A, a) → f
  5. Intersect(d, c) → D
  6. Intersect(f, a) → E
  7. Intersect(e, b) → F
  8. Polygon(D, E, F) → poly2, f_1, d_1, e_1
  9. Mirror(A, b) → A'
  10. Mirror(B, b) → B'
  11. Mirror(C, b) → C'
  12. Polygon(A', B', C') → poly1', c', a', b'
  13. Mirror(A, b) → A'_1
  14. Mirror(B, b) → B'_1
  15. Mirror(C, b) → C'_1
  16. Mirror(A, b) → A'_2
  17. Mirror(B, b) → B'_2
  18. Mirror(C, b) → C'_2
  19. Mirror(C, b') → C'_3
  20. Mirror(A, b') → A'_3
  21. Segment(C'_3, A'_3) → b'_3
  22. Mirror(A, b) → A'_4
  23. Mirror(B, b) → B'_3
  24. Mirror(C, b) → C'_4
  25. Mirror(A', a') → A''
  26. Mirror(B', a') → B''
  27. Mirror(C', a') → C''
  28. Mirror(A', c') → A''_1
  29. Mirror(B', c') → B''_1
  30. Mirror(C', c') → C''_1
  31. Polygon(A''_1, B''_1, C''_1) → poly1'', c''_1, a''_1, b''_1
  32. Mirror(A''_1, a''_1) → G
  33. Mirror(B''_1, a''_1) → H
  34. Mirror(C''_1, a''_1) → I
  35. Polygon(G, H, I) → poly1''', i, g, h
  36. Mirror(A, b) → A'_5
  37. Mirror(B, b) → B'_4
  38. Mirror(C, b) → C'_5
  39. Mirror(D, b) → D'
  40. Mirror(E, b) → E'
  41. Mirror(F, b) → F'
  42. Polygon(D', E', F') → poly2', f', d', e'
  43. Mirror(D', c') → D''
  44. Mirror(E', c') → E''
  45. Mirror(F', c') → F''
  46. Polygon(D'', E'', F'') → poly2'', f'', d'', e''
  47. Mirror(D'', g) → D'''
  48. Mirror(E'', g) → E'''
  49. Mirror(F'', g) → F'''
  50. Polygon(D''', E''', F''') → poly2''', f''', d''', e'''
  51. Mirror(G, h) → G'
  52. Mirror(H, h) → H'
  53. Mirror(I, h) → I'
  54. Polygon(G', H', I') → poly3, i', g', h'
  55. Mirror(D''', h) → J
  56. Mirror(E''', h) → K
  57. Mirror(F''', h) → L
  58. Polygon(J, K, L) → poly4, l, j, k
  59. Mirror(G', i') → G''
  60. Mirror(H', i') → H''
  61. Mirror(I', i') → I''
  62. Polygon(G'', H'', I'') → poly3', i'', g'', h''
  63. Mirror(J, i') → J'
  64. Mirror(K, i') → K'
  65. Mirror(L, i') → L'
  66. Polygon(J', K', L') → poly4', l', j', k'
  67. Mirror(G'', h'') → G'''
  68. Mirror(I'', h'') → I'''
  69. Mirror(G'', g'') → G'''_1
  70. Mirror(H'', g'') → H'''_1
  71. Mirror(I'', g'') → I'''_1
  72. Polygon(G'''_1, H'''_1, I'''_1) → poly3'', i'''_1, g'''_1, h'''_1
  73. Mirror(J', g'') → J''
  74. Mirror(K', g'') → K''
  75. Mirror(L', g'') → L''
  76. Polygon(J'', K'', L'') → poly4'', l'', j'', k''
  77. Segment(A, E) → n
  78. Segment(B, F) → p
  79. Segment(C, D) → q
  80. Intersect(a, f_1) → M
  81. Segment(M, L'') → r

AI Drawing Prompt Reference

Draw a triangle and construct perpendiculars from each vertex to the opposite side to form a new triangle. Reflect each vertex of the original triangle over the midpoint of the opposite side to create a new triangle. Continue reflecting the vertices of the new triangles over their respective sides to generate multiple nested triangles, highlighting geometric optimization and symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.