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Hexagonal Surprise!

Hexagonal Surprise!
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2017-08-06 10:57
PolygonTransformationGeometric Construction
This dynamic and modifiable illustration demonstrates phenomena that occur within any hexagon (convex or concave). It showcases various geometric phenomena within a hexagon. The example involves core mathematical concepts such as similar triangles and polygons. Using GeoGebra commands like Polygon, Segment, Midpoint, Line, etc., the construction of the hexagon and related geometric elements is demonstrated. Suitable for high school and above students or teachers.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D, E, F) → poly1, a, b, c, d, e, f
  2. Segment(F, A) → g
  3. Segment(A, B) → h
  4. Segment(B, C) → i
  5. Segment(C, D) → j
  6. Segment(D, E) → k
  7. Segment(E, F) → l
  8. Midpoint(l) → G
  9. Midpoint(g) → H
  10. Midpoint(h) → I
  11. Midpoint(i) → J
  12. Midpoint(j) → K
  13. Midpoint(k) → L
  14. Line(A, G) → m
  15. Line(E, H) → n
  16. Intersect(m, n) → M
  17. Line(F, L) → p
  18. Line(D, G) → q
  19. Intersect(p, q) → N
  20. Line(C, L) → r
  21. Line(E, K) → s
  22. Intersect(r, s) → O
  23. Line(D, J) → t
  24. Line(B, K) → f_1
  25. Intersect(t, f_1) → P
  26. Line(C, I) → g_1
  27. Line(A, J) → h_1
  28. Intersect(g_1, h_1) → Q
  29. Line(B, H) → i_1
  30. Line(F, I) → j_1
  31. Intersect(i_1, j_1) → R
  32. Polygon(M, N, O, P, Q, R) → poly2, m_1, n_1, o, p_1, q_1, r_1
  33. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], I) → A'
  34. Dilate(B, If[0 < u ≤ 1, 1 - u, 0], J) → B'
  35. Dilate(C, If[0 < u ≤ 1, 1 - u, 0], K) → C'
  36. Dilate(D, If[0 < u ≤ 1, 1 - u, 0], L) → D'
  37. Dilate(E, If[0 < u ≤ 1, 1 - u, 0], G) → E'
  38. Dilate(F, If[0 < u ≤ 1, 1 - u, 0], H) → F'
  39. Segment(A, A') → k_1
  40. Segment(B, B') → l_1
  41. Segment(C, C') → s_1
  42. Segment(D, D') → t_1
  43. Segment(E, E') → a_1
  44. Segment(F, F') → b_1
  45. Rotate(A, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], I) → A'_1
  46. Rotate(A', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], I) → A''
  47. Segment(A'_1, A'') → c_1
  48. Rotate(B, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], J) → B'_1
  49. Rotate(B', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], J) → B''
  50. Segment(B'_1, B'') → d_1
  51. Rotate(C, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], K) → C'_1
  52. Rotate(C', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], K) → C''
  53. Segment(C'_1, C'') → e_1
  54. Rotate(D, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], L) → D'_1
  55. Rotate(D', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], L) → D''
  56. Segment(D'_1, D'') → f_2
  57. Rotate(E, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], G) → E'_1
  58. Rotate(E', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], G) → E''
  59. Segment(E'_1, E'') → g_2
  60. Rotate(F, If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], H) → F'_1
  61. Rotate(F', If[1 < u ≤ 2, ((-(u - 1)) * π), (-π)], H) → F''
  62. Segment(F'_1, F'') → h_2
  63. Dilate(A, If[2 < u ≤ 3, 3 - u, 0], G) → A'_2
  64. Dilate(E, If[2 < u ≤ 3, 3 - u, 0], H) → E'_2
  65. Segment(A, A'_2) → i_2
  66. Segment(E, E'_2) → j_2
  67. Dilate(F, If[3 < u ≤ 4, 4 - u, 0], L) → F'_2
  68. Dilate(D, If[3 < u ≤ 4, 4 - u, 0], G) → D'_2
  69. Segment(F, F'_2) → k_2
  70. Segment(D, D'_2) → l_2
  71. Dilate(C, If[4 < u ≤ 5, 5 - u, 0], L) → C'_2
  72. Dilate(E, If[4 < u ≤ 5, 5 - u, 0], K) → E'_3
  73. Segment(C, C'_2) → m_2
  74. Segment(E, E'_3) → n_2
  75. Dilate(D, If[5 < u ≤ 6, 6 - u, 0], J) → D'_3
  76. Dilate(B, If[5 < u ≤ 6, 6 - u, 0], K) → B'_2
  77. Segment(B, B'_2) → p_2
  78. Segment(D, D'_3) → q_2
  79. Dilate(C, If[6 < u ≤ 7, 7 - u, 0], I) → C'_3
  80. Dilate(A, If[6 < u ≤ 7, 7 - u, 0], J) → A'_3
  81. Segment(A, A'_3) → r_2
  82. Segment(C, C'_3) → s_2
  83. Dilate(B, If[7 < u ≤ 8, 8 - u, 0], H) → B'_3
  84. Dilate(F, If[7 < u ≤ 8, 8 - u, 0], I) → F'_3
  85. Segment(F, F'_3) → t_2
  86. Segment(B, B'_3) → a_2
  87. Dilate(R, If[15 < u ≤ 16, 16 - u, 0], M) → R'
  88. Segment(R, R') → b_2
  89. Dilate(R, If[16 < u ≤ 17, 17 - u, 0], P) → R'_1
  90. Vector(R, R'_1) → v
  91. Translate(R, v) → R'_2
  92. Translate(R', v) → R''
  93. Segment(R'_2, R'') → c_2
  94. Dilate(M, If[17 < u ≤ 18, 18 - u, 0], N) → M'
  95. Segment(M, M') → d_2
  96. Dilate(M, If[18 < u ≤ 19, 19 - u, 0], Q) → M'_1
  97. Vector(M, M'_1) → w
  98. Translate(M, w) → M'_2
  99. Translate(M', w) → M''
  100. Segment(M'_2, M'') → e_2
  101. Dilate(N, If[19 < u ≤ 20, 20 - u, 0], O) → N'
  102. Segment(N, N') → f_3
  103. Dilate(N, If[20 < u ≤ 21, 21 - u, 0], R) → N'_1
  104. Vector(N, N'_1) → u_1
  105. Translate(N, u_1) → N'_2
  106. Translate(N', u_1) → N''
  107. Segment(N'_2, N'') → g_3