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Dynamic Demonstration of Concurrent Relationships in Any Hexagon

Dynamic Demonstration of Concurrent Relationships in Any Hexagon
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2017-05-16 02:23
PolygonVectorGeometric Construction
This example dynamically illustrates concurrent relationships within any hexagon. It involves core math concepts like polygons, vectors, and geometric transformations (rotation, dilation). Construction highlights include Midpoint, Polygon, Dilate, Segment, Rotate, and Intersect commands. Suitable for high school and above, ideal for exploring complex geometric properties.

GeoGebra Commands & Parameters

  1. Midpoint(D, E) → B_2
  2. Midpoint(F, A) → V_1
  3. Midpoint(B_2, V_1) → M
  4. Midpoint(A, B) → W_1
  5. Midpoint(E, F) → U_1
  6. Midpoint(C, D) → A_2
  7. Midpoint(B, C) → Z_1
  8. Polygon(A, B, C, D, E, F) → poly1, a, b, c, d, e, f
  9. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], W_1) → A'
  10. Segment(A, A') → k_1
  11. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], Z_1) → B'
  12. Segment(B, B') → l_1
  13. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], A_2) → C'
  14. Segment(C, C') → m_1
  15. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], B_2) → D'
  16. Segment(D, D') → n_1
  17. Dilate(E, If[0 < o ≤ 1, 1 - o, 0], U_1) → E'
  18. Segment(E, E') → p_1
  19. Dilate(F, If[0 < o ≤ 1, 1 - o, 0], V_1) → F'
  20. Segment(F, F') → q_1
  21. Rotate(C, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], C) → C'_1
  22. Rotate(C', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], C) → C''
  23. Rotate(D, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], D) → D'_1
  24. Rotate(D', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], D) → D''
  25. Rotate(E, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], E) → E'_1
  26. Rotate(E', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], E) → E''
  27. Rotate(F, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], F) → F'_1
  28. Rotate(F', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], F) → F''
  29. Rotate(A, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], A) → A'_1
  30. Rotate(A', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], A) → A''
  31. Rotate(B, If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], B) → B'_1
  32. Rotate(B', If[1.5 < o ≤ 2.5, ((-(o - 1.5)) * π / 3), ((-π)) / 3], B) → B''
  33. Rotate(C, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], D) → C'_2
  34. Rotate(C', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], D) → C''_1
  35. Rotate(D, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], E) → D'_2
  36. Rotate(D', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], E) → D''_1
  37. Rotate(E, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], F) → E'_2
  38. Rotate(E', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], F) → E''_1
  39. Rotate(F, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], A) → F'_2
  40. Rotate(F', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], A) → F''_1
  41. Rotate(A, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], B) → A'_2
  42. Rotate(A', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], B) → A''_1
  43. Rotate(B, If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], C) → B'_2
  44. Rotate(B', If[2.5 < o ≤ 3.5, ((o - 2.5) * π / 3), π / 3], C) → B''_1
  45. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], W_1) → A'_6
  46. Rotate(A', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], W_1) → A''_2
  47. Segment(A'_6, A''_2) → r_1
  48. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], Z_1) → B'_6
  49. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], Z_1) → B''_2
  50. Segment(B'_6, B''_2) → s_1
  51. Rotate(C, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], A_2) → C'_7
  52. Rotate(C', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], A_2) → C''_2
  53. Segment(C'_7, C''_2) → t_1
  54. Rotate(D, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], B_2) → D'_5
  55. Rotate(D', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], B_2) → D''_2
  56. Segment(D'_5, D''_2) → a_1
  57. Rotate(E, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], U_1) → E'_6
  58. Rotate(E', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], U_1) → E''_2
  59. Segment(E'_6, E''_2) → b_1
  60. Rotate(F, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], V_1) → F'_6
  61. Rotate(F', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], V_1) → F''_2
  62. Segment(F'_6, F''_2) → c_1
  63. Midpoint(Z_1, B_2) → E_2
  64. Dilate(Z_1, If[2 < o ≤ 3, 3 - o, 0], E_2) → G
  65. Segment(Z_1, G) → g
  66. Dilate(B_2, If[2 < o ≤ 3, 3 - o, 0], M) → H
  67. Segment(B_2, H) → h
  68. Rotate(Z_1, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], E_2) → I
  69. Rotate(G, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], E_2) → G'
  70. Segment(I, G') → g'
  71. Rotate(B_2, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], M) → N
  72. Rotate(H, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], M) → H'_1
  73. Segment(N, H'_1) → h'
  74. Midpoint(V_1, Z_1) → J
  75. Dilate(V_1, If[2 < o ≤ 3, 3 - o, 0], J) → K
  76. Segment(V_1, K) → i
  77. Rotate(V_1, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], J) → L
  78. Rotate(K, If[3 < o ≤ 4, ((-(o - 3)) * π), (-π)], J) → K'
  79. Segment(L, K') → i'
  80. Polygon(V_1, Z_1, B_2) → poly2, b_2, v_1, z_1
  81. Midpoint(W_1, A_2) → O
  82. Midpoint(A_2, U_1) → P
  83. Midpoint(U_1, W_1) → Q
  84. Dilate(W_1, If[4 < o ≤ 5, 5 - o, 0], O) → R
  85. Dilate(A_2, If[4 < o ≤ 5, 5 - o, 0], P) → S
  86. Dilate(U_1, If[4 < o ≤ 5, 5 - o, 0], Q) → T
  87. Segment(W_1, R) → j
  88. Segment(A_2, S) → k
  89. Segment(U_1, T) → l
  90. Rotate(A_2, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], P) → U
  91. Rotate(S, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], P) → S'
  92. Segment(U, S') → k'
  93. Rotate(W_1, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], O) → V
  94. Rotate(R, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], O) → R'
  95. Segment(V, R') → j'
  96. Rotate(U_1, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], Q) → W
  97. Rotate(T, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], Q) → T'
  98. Segment(W, T') → l'
  99. Polygon(W_1, A_2, U_1) → poly3, u_1, w_1, a_2
  100. Segment(U_1, O) → m
  101. Segment(W_1, P) → n
  102. Intersect(m, n) → Z
  103. Dilate(B_2, If[6 < o ≤ 7, 7 - o, 0], J) → A_1
  104. Segment(B_2, A_1) → p
  105. Dilate(V_1, If[7 < o ≤ 8, 8 - o, 0], Z) → B_1
  106. Dilate(Z, If[8 < o ≤ 9, 9 - o, 0], Q) → Z'
  107. Segment(V_1, B_1) → q
  108. Dilate(Z, If[8 < o ≤ 9, 9 - o, 0], E_2) → Z'_1
  109. Segment(Z, Z'_1) → r
  110. Dilate(U_1, If[9 < o ≤ 10, 10 - o, 0], Z) → C_1
  111. Segment(U_1, C_1) → s
  112. Dilate(Z, If[10 < o ≤ 11, 11 - o, 0], O) → Z'_2
  113. Segment(ZZ, Z'_2) → t
  114. Dilate(W_1, If[11 < o ≤ 12, 12 - o, 0], Z) → D_1
  115. Segment(W_1, D_1) → f_1
  116. Dilate(Z, If[12 < o ≤ 13, 13 - o, 0], P) → Z'_3
  117. Segment(Z, Z'_3) → g_1