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Dynamic Relationships in Two Squares Off Any Triangle's Sides

Dynamic Relationships in Two Squares Off Any Triangle's Sides
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2017-04-21 00:59
TrianglePolygonVectorTransformationGeometric Construction
This example demonstrates surprising relationships within two squares constructed off any type of triangle's sides. It involves core mathematical concepts like geometric construction, angle relationships, vector operations, and dynamic transformations. The construction utilizes commands and objects such as Rotate, Ray, Point, Segment, Midpoint, Polygon, Angle, Min, Circle, Intersect, Line, CircleSector, Dilate, Vector, and Translate from GeoGebra. It is suitable for high school and above, helping learners understand invariance principles in dynamic geometry and transformations.

GeoGebra Commands & Parameters

  1. Rotate(A, (a * π / 180), B) → A'
  2. Ray(B, A') → g
  3. Point(g) → C
  4. Rotate(A, If[0 < u ≤ 1, (u * π / 2), π / 2], C) → A'_1
  5. Rotate(C, If[0 < u ≤ 1, (u * π / 2), π / 2], C) → C'
  6. Segment(C', A'_1) → i'
  7. Midpoint(i') → V
  8. Polygon(C, A, 4) → poly1, j, k, l, m, D, E
  9. Angle(A, B, A') → α
  10. Segment(A, B) → f
  11. Segment(B, C) → h
  12. Segment(C, A) → i
  13. Polygon(B, C, 4) → poly2, n, p, q, r, F, G
  14. Angle(poly2) → β, γ, δ, ε
  15. Angle(poly1) → ζ, η, θ, ι
  16. Segment(C, E) → s
  17. Segment(E, D) → t
  18. Segment(D, A) → b
  19. Segment(B, G) → c
  20. Segment(G, F) → d
  21. Segment(F, C) → e
  22. Min(i, h) → o
  23. Circle(E, (0.15 * o)) → c_1
  24. Polygon(B, A, C) → poly3, c_2, b_1, a_1
  25. Intersect(c_1, s, 1) → H
  26. Intersect(c_1, t, 1) → I
  27. Line(B, E) → f_1
  28. Line(A, F) → g_1
  29. Line(G, D) → h_1
  30. Intersect(g_1, f_1) → J
  31. Circle(A, (0.15 * o)) → d_1
  32. Intersect(d_1, b, 1) → K
  33. Intersect(d_1, i, 1) → L
  34. CircleSector(A, K, L) → e_1
  35. Rotate(B, If[0 < u ≤ 1, (u * π / 2), π / 2], B) → B'
  36. Rotate(C, If[0 < u ≤ 1, (u * π / 2), π / 2], B) → C'_1
  37. Segment(B', C'_1) → h'
  38. Rotate(B', If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], G) → B''
  39. Rotate(C'_1, If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], G) → M
  40. Segment(B'', M) → h''
  41. Rotate(C', If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], E) → C''
  42. Rotate(A'_1, If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], E) → N
  43. Segment(C'', N) → i''
  44. Rotate(C'', If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], D) → C'''
  45. Rotate(N, If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], D) → N'
  46. Segment(C''', N') → i'''
  47. Rotate(B'', If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], F) → B'''
  48. Rotate(M, If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], F) → M'
  49. Segment(B''', M') → h'''
  50. Dilate(A, If[3.5 < u ≤ 4.5, 4.5 - u, 0], D) → A'_2
  51. Vector(A, A'_2) → v
  52. Translate(e_1, v) → k_1
  53. Dilate(A, If[3.5 < u ≤ 4.5, 4.5 - u, 0], C) → A'_3
  54. Vector(A, A'_3) → w
  55. Translate(e_1, w) → p_1
  56. Line(C, A) → i_1
  57. Point(i_1) → O
  58. Angle(O, C, F) → κ
  59. Rotate(p_1, If[4.5 < u ≤ 5.5, ((u - 4.5) * (π / 2 + κ)), π / 2 + κ], C) → q_1
  60. Rotate(k_1, If[4.5 < u ≤ 5.5, ((u - 4.5) * π / 2), π / 2], D) → r_1
  61. Dilate(D, If[5.5 < u ≤ 6.5, 6.5 - u, 0], E) → D'
  62. Dilate(C, If[5.5 < u ≤ 6.5, 6.5 - u, 0], F) → C'_2
  63. Vector(C, C'_2) → u_1
  64. Vector(D, D') → v_1
  65. Translate(r_1, v_1) → s_1
  66. Translate(q_1, u_1) → t_1
  67. Rotate(s_1, If[6.5 < u ≤ 7.5, ((u - 6.5) * π / 2), π / 2], E) → d_2
  68. Rotate(t_1, If[6.5 < u ≤ 7.5, ((u - 6.5) * π / 2), π / 2], F) → e_2
  69. Dilate(F, If[7.5 < u ≤ 8.5, 8.5 - u, 0], G) → F'
  70. Dilate(E, If[7.5 < u ≤ 8.5, 8.5 - u, 0], V) → E'
  71. Vector(E, E') → w_1
  72. Vector(F, F') → j_1
  73. Translate(d_2, w_1) → f_2
  74. Translate(e_2, j_1) → g_2
  75. Rotate(g_2, If[8.5 < u ≤ 9.5, ((u - 8.5) * π / 2), π / 2], G) → h_2
  76. Circle(G, 2) → p_2
  77. Circle(D, 2) → q_2
  78. Circle(F, 2) → r_2
  79. Intersect(r_2, g_1, 2) → P
  80. Intersect(p_2, h_1, 1) → Q
  81. Intersect(d_1, g_1, 1) → R
  82. Intersect(q_2, h_1, 2) → S
  83. Ray(D, S) → l_1
  84. Ray(G, Q) → m_1
  85. Ray(F, P) → n_1
  86. Ray(A, R) → i_2
  87. Circle(B, 2) → s_2
  88. Intersect(s_2, f_1, 1) → T
  89. Intersect(c_1, f_1, 2) → U
  90. Ray(B, T) → j_2
  91. Ray(E, U) → l_2
  92. Rotate(f_2, If[8.5 < u ≤ 9.5, ((u - 8.5) * π / 2), π / 2], V) → k_2
  93. Dilate(V, If[9.5 < u ≤ 10.5, 10.5 - u, 0], C) → V'
  94. Vector(V, V') → u_2
  95. Translate(k_2, u_2) → t_2
  96. Dilate(B, If[11 < u ≤ 12, 12 - u, 0], E) → B'_1
  97. Segment(B, B'_1) → m_2
  98. Dilate(A, If[12 < u ≤ 13, 13 - u, 0], J) → A'_4
  99. Segment(A, A'_4) → n_2
  100. Dilate(J, If[13 < u ≤ 14, 14 - u, 0], F) → J'
  101. Segment(J, J') → a_2
  102. Dilate(G, If[14 < u ≤ 15, 15 - u, 0], J) → G'
  103. Segment(G, G') → b_2
  104. Circle(J, (0.15 * o)) → c_3
  105. Intersect(c_3, g_1, 1) → W
  106. Intersect(c_3, h_1, 2) → Z
  107. Sector(c_3, W, Z) → d_3
  108. Dilate(JJ, If[15 < u ≤ 16, 16 - u, 0], D) → JJ'
  109. Segment(JJ, JJ') → f_3
  110. Line(G, n_2) → g_3
  111. Point(g_3) → A_1
  112. Line(G, j_2) → h_3
  113. Angle(F, G, A_1) → λ
  114. Rotate(h_2, If[16 < u ≤ 17, ((u - 16) * λ), λ], G) → e_3
  115. Dilate(G, If[17 < u ≤ 18, 18 - u, 0], JJ) → G'_1
  116. Vector(G, G'_1) → v_2
  117. Translate(e_3, v_2) → k_3
  118. Angle(A, JJ, D) → μ
  119. Rotate(W, If[18 < u ≤ 19, ((u - 18) * π / 4), π / 4], JJ) → W'
  120. CircleSector(JJ, W, W') → p_3
  121. Rotate(p_3, If[19.5 < u ≤ 20.5, ((u - 19.5) * π / 4), π / 4], JJ) → q_3
  122. Circle(B, (0.15 * o)) → r_3
  123. Intersect(r_3, f, 1) → B_1
  124. Intersect(r_3, h, 1) → C_1
  125. CircleSector(B, B_1, C_1) → s_3