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Dynamic Relationship of a Right-Triangle Problem

Dynamic Relationship of a Right-Triangle Problem
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2017-05-03 00:47
TriangleCircleVectorGeometric ConstructionSlider Animation
This example demonstrates the dynamic relationship of a right-triangle problem set posted by Antonio Gutierrez on Twitter. It showcases important properties and relationships of right triangles. Key mathematical concepts include properties of right triangles, similar triangles, midpoints, and circle properties. Using various GeoGebra commands such as semicircle, point, segment, midpoint, orthogonal line, circle, intersect, circle sector, dilate, vector, polygon, rotate, ray, line, translate, etc., dynamic exploration of these properties is possible. Suitable for middle and high school students learning geometric construction and dynamic geometry.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → d
  2. Point(d) → C
  3. Segment(C, B) → f
  4. Segment(A, C) → e
  5. Segment(A, B) → a
  6. Min(a, f) → m
  7. Min(a, e) → l
  8. Min(l, m) → n
  9. Midpoint(a) → C_1
  10. OrthogonalLine(C_1, a) → b
  11. Circle(C_1, A) → c
  12. OrthogonalLine(C, a) → g
  13. Intersect(g, a) → E
  14. Segment(C, E) → h
  15. Circle(A, (0.2 * n)) → k
  16. Circle(C, (0.2 * n)) → p
  17. Circle(B, (0.2 * n)) → q
  18. Intersect(p, h, 1) → I
  19. Intersect(p, f, 1) → J
  20. Intersect(q, a, 1) → K
  21. Intersect(q, f, 1) → L
  22. Intersect(k, a, 1) → M
  23. Intersect(k, e, 1) → N
  24. CircleSector(A, M, N) → r
  25. Dilate(C, If[-2 < i < -1, -1 - i, 0], E) → D'
  26. Dilate(A, If[-0.5 < i ≤ 0.5, 0.5 - i, 0], C) → A'
  27. Vector(A, A') → u
  28. Dilate(B, If[1.5 < i ≤ 2.5, 2.5 - i, 0], C) → B'
  29. Vector(B, B') → v
  30. Polygon(A, B, C) → poly3, d_3, a_2, b_2
  31. Angle(C, B, A) → δ
  32. Rotate(C, δ / 2, B) → C'
  33. Angle(C, B, C') → ε
  34. Ray(B, C') → h_1
  35. Intersect(e, h_1) → D
  36. Angle(B, A, C) → ζ
  37. Rotate(B, ζ / 2, A) → B'_1
  38. Angle(B, A, B'_1) → η
  39. Ray(A, B'_1) → i_1
  40. Intersect(f, i_1) → O
  41. Intersect(h_1, i_1) → P
  42. Intersect(q, h_1, 1) → Q
  43. Intersect(q, f, 1) → R
  44. Intersect(k, a, 1) → S
  45. Intersect(k, i_1, 1) → T
  46. CircleSector(A, S, T) → c_1
  47. CircleSector(B, R, Q) → k_1
  48. Polygon(A, B, C) → poly1, c_2, a_1, b_1
  49. OrthogonalLine(P, a) → j
  50. Intersect(a, j) → F
  51. Angle(P, F, A) → α
  52. Line(B, C) → j_1
  53. Circle(C, 1) → d_1
  54. Intersect(d_1, j_1, 2) → G
  55. Ray(C, G) → l_1
  56. Circle(C, (0.5 * Radius[k])) → e_1
  57. Intersect(e_1, e, 1) → H
  58. Intersect(e_1, l_1, 1) → U
  59. CircleSector(C, U, H) → p_1
  60. Rotate(p_1, If[0 < i ≤ 1, (i * π / 2), π / 2], C) → s
  61. Dilate(A, If[1 < i ≤ 2, 2 - i, 0], P) → A'_1
  62. Dilate(B, If[1 < i ≤ 2, 2 - i, 0], P) → B'_2
  63. Segment(A, T) → t
  64. Segment(B, B'_2) → f_1
  65. Segment(A, A'_1) → g_1
  66. OrthogonalLine(P, g_1) → m_1
  67. Line(C, m_1) → n_1
  68. Line(C, g_1) → q_1
  69. Rotate(k_1, If[2.5 < i ≤ 3.5, ((i - 2.5) * ε), ε], B) → r_1
  70. Rotate(c_1, If[3.5 < i ≤ 4.5, ((i - 3.5) * ζ / 2), ζ / 2], A) → s_1
  71. Intersect(m_1, a) → V
  72. Dilate(P, If[4.5 < i ≤ 5.5, 5.5 - i, 0], V) → P'
  73. Segment(P, P') → t_1
  74. Dilate(C, If[5.5 < i ≤ 6.5, 6.5 - i, 0], P) → C'_1
  75. Vector(C, C'_1) → u_1
  76. Translate(s, u_1) → s'
  77. Point(q_1) → W
  78. Angle(W, C, A) → β
  79. Rotate(s', If[6.5 < i ≤ 7.5, ((-(i - 6.5)) * β), (-β)], P) → s''
  80. Segment(C, B) → g_2
  81. OrthogonalLine(V, l_1) → h_2
  82. Line(C, B) → i_2
  83. Intersect(i_2, h_2) → Z
  84. Dilate(V, If[7.5 < i ≤ 8.5, 8.5 - i, 0], Z) → V'
  85. Segment(V, V') → j_2
  86. Dilate(C, If[8.5 < i ≤ 9.5, 9.5 - i, 0], Z) → C'_2
  87. Vector(C, C'_2) → v_1
  88. Translate(s, v_1) → s'_1
  89. Line(A, C) → k_2
  90. OrthogonalLine(P, k_2) → l_2
  91. Intersect(k_2, l_2) → A_1
  92. Dilate(P, If[9.5 < i ≤ 10.5, 10.5 - i, 0], A_1) → P'_1
  93. Segment(P, P'_1) → f_2
  94. Dilate(C, If[10.5 < i ≤ 11.5, 11.5 - i, 0], A_1) → C'_3
  95. Vector(C, C'_3) → w_1
  96. Translate(s, w_1) → s'_2
  97. Dilate(A_1, If[11.5 < i ≤ 12.5, 12.5 - i, 0], C) → B_1
  98. Vector(A_1, B_1) → u_2
  99. Translate(P, u_2) → P'_2
  100. Translate(P'_1, u_2) → D_1
  101. Segment(P'_2, D_1) → m_2
  102. Dilate(A_1, If[12.5 < i ≤ 13.5, 13.5 - i, 0], P'_2) → E_1
  103. Vector(A_1, E_1) → v_2
  104. Translate(P, v_2) → P'_3
  105. Translate(P'_1, v_2) → F_1
  106. Segment(P'_3, F_1) → n_2