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Cool Phenomenon in a Parallelogram

Cool Phenomenon in a Parallelogram
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2017-09-07 11:31
PolygonTransformationGeometric Construction
This example demonstrates a dynamic and modifiable relationship that occurs when additional parallel segments are added to the sides of a parallelogram. Key mathematical concepts include parallel lines, equilateral triangles, and integers. The construction uses GeoGebra commands such as Line, Rotate, Segment, etc. Suitable for middle school and above students to learn geometric construction and transformations.

GeoGebra Commands & Parameters

  1. Line(A, B) → r_2
  2. Rotate(B, a°, A) → B'
  3. Ray(A, B') → g
  4. Line(B, g) → h
  5. Point(g) → C
  6. Segment(A, C) → p_1
  7. Point(p_1) → T
  8. Ray(A, B) → f
  9. Line(C, f) → i
  10. Intersect(h, i) → D
  11. Segment(C, D) → q_1
  12. Line(T, q_1) → t_1
  13. Point(q_1) → U
  14. Line(U, p_1) → s_1
  15. Intersect(t_1, s_1) → Z
  16. Line(C, Z) → g_2
  17. Segment(T, C) → j_1
  18. Segment(A, T) → i_1
  19. Segment(U, D) → l_1
  20. Segment(C, U) → k_1
  21. Angle(B, A, B') → α
  22. Segment(C, D) → j
  23. Segment(A, B) → k
  24. Segment(B, D) → l
  25. Segment(A, C) → m
  26. Min({i_1, j_1, k_1, l_1}) → b
  27. Circle(C, (0.3 * b)) → d
  28. Circle(A, (0.35 * b)) → e
  29. Circle(B, (0.4 * b)) → p
  30. Circle(D, (0.3 * b)) → c
  31. Intersect(e, k, 1) → E
  32. Intersect(e, m, 1) → F
  33. Intersect(p, k, 1) → G
  34. Intersect(p, l, 1) → H
  35. Intersect(d, g, 2) → I
  36. Ray(C, I) → q
  37. Intersect(c, h, 2) → J
  38. Ray(D, J) → r
  39. Intersect(c, i, 2) → K
  40. Ray(D, K) → s
  41. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  42. Segment(A, A') → t
  43. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  44. CircleSector(A, E, E') → c_1
  45. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
  46. Vector(A, A'_1) → u
  47. Translate(c_1, u) → d_1
  48. Segment(A, A'_1) → f_1
  49. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  50. Vector(C, C') → v
  51. Translate(d_1, v) → e_1
  52. Segment(C, C') → g_1
  53. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  54. Segment(D, D') → h_1
  55. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  56. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  57. Vector(A, A'_2) → w
  58. Translate(A, w) → A'_3
  59. Translate(A', w) → A''
  60. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  61. Vector(A, A'_4) → u_1
  62. Translate(A, u_1) → A'_5
  63. Translate(A'_1, u_1) → L
  64. Intersect(d, i) → M, N
  65. Intersect(d, g) → O, P
  66. Angle(A, C, N) → β
  67. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  68. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  69. Vector(C, C'_1) → v_1
  70. Segment(D, B) → r_1
  71. Segment(A, B) → f_2
  72. Intersect(f_2, s_1) → V
  73. Intersect(t_1, r_1) → W
  74. Segment(D, V) → i_2
  75. Segment(A, W) → j_2
  76. Intersect(g_2, r_2) → A_1
  77. Dilate(T, If[6.5 < o ≤ 7.5, 7.5 - o, 0], W) → T'
  78. Segment(T, T') → m_1
  79. Dilate(U, If[7.5 < o ≤ 8.5, 8.5 - o, 0], Z) → U'
  80. Segment(U, U') → n_1
  81. Dilate(Z, If[8.5 < o ≤ 9.5, 9.5 - o, 0], V) → Z'
  82. Segment(Z, Z') → h_2
  83. Dilate(A, If[9.5 < o ≤ 10.5, 10.5 - o, 0], T) → A'_6
  84. Vector(A, A'_6) → w_1
  85. Translate(c_1, w_1) → e_2
  86. Dilate(D, If[10.5 < o ≤ 11.5, 11.5 - o, 0], U) → C'_2
  87. Vector(D, C'_2) → u_2
  88. Dilate(T, If[10.5 < o ≤ 11.5, 11.5 - o, 0], Z) → T'_1
  89. Vector(T, T'_1) → v_2
  90. Translate(e_2, v_2) → k_2
  91. Intersect(g_2, j_2) → Q
  92. Dilate(C, If[11.5 < o ≤ 12.5, 12.5 - o, 0], A_1) → C'_3
  93. Segment(C, C'_3) → l_2
  94. Dilate(A, If[12.5 < o ≤ 13.5, 13.5 - o, 0], Q) → A'_7
  95. Segment(A, A'_7) → m_2
  96. Dilate(Q, If[13.5 < o ≤ 14.5, 14.5 - o, 0], W) → Q'
  97. Segment(Q, Q') → n_2
  98. Dilate(D, If[14.5 < o ≤ 15.5, 15.5 - o, 0], Q) → D'_1
  99. Segment(D, D'_1) → p_2
  100. Intersect(l_2, m_2) → R
  101. Dilate(Q, If[15.5 < o ≤ 16.5, 16.5 - o, 0], V) → Q'_1
  102. Segment(Q, Q'_1) → q_2