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Dynamic Equilateral Triangle & Median Value Problem

Dynamic Equilateral Triangle & Median Value Problem
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2018-07-04 11:00
TriangleFunctionGeometric Construction
This example dynamically illustrates a problem posted by Antonio Gutierrez on Twitter involving equilateral triangles, median values, exponential functions, area, cosine, and rhombuses. It covers core math concepts like equilateral triangle construction, median properties, exponential function graphs, area calculation, cosine theorem, and rhombus characteristics. Construction highlights include commands like Polygon, Line, OrthogonalLine, Intersect, Midpoint, Segment, Point, Rotate, Angle, Ray, Dilate, Min, Circle, CircleSector, Vector, and Translate. Suitable for high school mathematics education to explore geometric transformations combined with function problems.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Line(C, A) → k
  3. OrthogonalLine(A, g) → i
  4. OrthogonalLine(B, h) → j
  5. Intersect(i, j) → D
  6. Intersect(h, j) → E
  7. Midpoint(C, B) → F
  8. Segment(C, F) → l
  9. Point(l) → G
  10. Rotate(G, 60°, D) → G'
  11. Angle(G, D, G') → α
  12. Ray(D, G') → m
  13. Intersect(h, m) → H
  14. Polygon(H, G, C) → t1, c, h_1, g_1
  15. Polygon(A, B, G, H) → q1, a_1, b, g_2, h_2
  16. Angle(A, F, B) → β
  17. Angle(A, E, B) → γ
  18. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  19. Segment(A, A') → n
  20. Rotate(A, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], A) → A'_1
  21. Rotate(A', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], A) → A''
  22. Segment(A'_1, A'') → n'
  23. Rotate(A'_1, If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], C) → I
  24. Rotate(A'', If[2 < d ≤ 3, ((d - 2) * π / 3), π / 3], C) → A'''
  25. Segment(I, A''') → n''
  26. Dilate(A, If[3 < d ≤ 4, 4 - d, 0], F) → J
  27. Segment(A, J) → p
  28. Dilate(B, If[4 < d ≤ 5, 5 - d, 0], D) → B'
  29. Segment(B, B') → q
  30. Dilate(D, If[5 < d ≤ 6, 6 - d, 0], E) → D'
  31. Segment(D, D') → r
  32. Segment(C, A) → s
  33. Segment(D, G) → t
  34. Segment(D, H) → e
  35. Segment(C, G) → f_1
  36. Segment(C, H) → i_1
  37. Min({e, t, f_1, i_1}) → o
  38. Circle(C, (0.45 * o)) → c_1
  39. Intersect(s, i_1) → K
  40. Intersect(c_1, i_1, 1) → L
  41. Rotate(L, If[9 < d ≤ 10, ((d - 9) * π / 3), π / 3], C) → L'
  42. CircleSector(C, L, L') → d_1
  43. Dilate(D, If[9 < d ≤ 10, 10 - d, 0], G) → D'_1
  44. Dilate(D, If[9 < d ≤ 10, 10 - d, 0], H) → D'_2
  45. Segment(D, D'_1) → j_1
  46. Segment(D, D'_2) → k_1
  47. Line(C, j_1) → l_1
  48. Line(C, k_1) → m_1
  49. Point(l_1) → M
  50. Angle(L, C, M) → δ
  51. Dilate(C, If[11 < d ≤ 12, 12 - d, 0], D) → C'
  52. Vector(C, C') → u
  53. Translate(d_1, u) → e_1
  54. Rotate(e_1, If[11 < d ≤ 12, ((d - 11) * δ), δ], C') → p_1
  55. Dilate(C, If[12 < d ≤ 13, 13 - d, 0], H) → C'_1
  56. Dilate(G, If[12 < d ≤ 13, 13 - d, 0], C) → C'_2
  57. Dilate(H, If[12 < d ≤ 13, 13 - d, 0], G) → H'
  58. Segment(H, H') → n_1
  59. Segment(G, C'_2) → q_1
  60. Segment(C, C'_1) → r_1
  61. Polygon(H, G, C) → t2, c_2, h_3, g_3
  62. Angle(G, H, C) → ε
  63. Line(A, B) → s_1
  64. Line(G, s_1) → t_1
  65. Point(t_1) → N
  66. Angle(H, G, N) → ζ
  67. Dilate(H, If[14 < d ≤ 15, 15 - d, 0], A) → H'_1
  68. Vector(H, H'_1) → v
  69. Translate(H, v) → H'_2
  70. Translate(H, v) → H'_3
  71. Translate(H', v) → H''
  72. Segment(H'_3, H'') → b_1
  73. Rotate(H'_3, If[14 < d ≤ 15, ((d - 14) * ζ), ζ], H'_1) → O
  74. Rotate(H'', If[14 < d ≤ 15, ((d - 14) * ζ), ζ], H'_1) → H'''
  75. Segment(O, H''') → f_2
  76. Dilate(H, If[15 < d ≤ 16, 16 - d, 0], H''') → H'_4
  77. Vector(H, H'_4) → w
  78. Translate(C, w) → C'_3
  79. Translate(C'_1, w) → P
  80. Segment(C'_3, P) → i_2
  81. Rotate(C'_3, If[15 < d ≤ 16, ((-(d - 15)) * π / 3), ((-π)) / 3], P) → Q
  82. Rotate(P, If[15 < d ≤ 16, ((-(d - 15)) * π / 3), ((-π)) / 3], P) → P'
  83. Segment(Q, P') → j_2
  84. Dilate(C, If[16 < d ≤ 17, 17 - d, 0], Q) → G'_1
  85. Vector(C, G'_1) → u_1
  86. Translate(G, u_1) → G'_2
  87. Translate(C'_2, u_1) → R
  88. Segment(G'_2, R) → k_2
  89. Rotate(G'_2, If[16 < d ≤ 17, ((d - 16) * π / 3), π / 3], R) → S
  90. Rotate(R, If[16 < d ≤ 17, ((d - 16) * π / 3), π / 3], R) → R'
  91. Segment(S, R') → l_2