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Dynamic Illustration of a Circle Tangent to the Hypotenuse of an Isosceles Right

Dynamic Illustration of a Circle Tangent to the Hypotenuse of an Isosceles Right
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2017-07-04 11:20
TriangleCircleGeometric Construction
This example dynamically illustrates the relationship between the circumcircle of an isosceles right triangle and its hypotenuse, which is tangent to the circle. Key math concepts include circle, isosceles triangle, tangent line, trigonometric functions, and geometric constructions. The construction utilizes GeoGebra commands such as Circle, Segment, OrthogonalLine, Rotate, Angle, Ray, Intersect, Polygon, Line, Point, Dilate, CircleSector, Vector, CircleArc, Midpoint, and Translate. It is suitable for high school mathematics education, helping students understand the properties of circles and triangles, tangent lines, and dynamic geometric relationships.

GeoGebra Commands & Parameters

  1. Circle(B, C) → c
  2. Segment(B, C) → a
  3. OrthogonalLine(C, a) → b
  4. Rotate(C, 45°, B) → C'
  5. Angle(C, B, C') → α
  6. Ray(B, C') → d
  7. Intersect(b, d) → A
  8. Segment(A, C) → e
  9. Segment(B, A) → f
  10. Polygon(B, C, A) → poly1, a_1, b_1, c_1
  11. Line(B, C) → h
  12. Point(h) → D
  13. Point(h) → E
  14. Ray(C, E) → i
  15. Point(i) → F
  16. Ray(C, F) → j
  17. Dilate(B, If[0 < g ≤ 1, 1 - g, 0], C) → B'
  18. Segment(B, B') → k
  19. Dilate(C, If[1 < g ≤ 2, 2 - g, 0], A) → C'_1
  20. Segment(C, C'_1) → l
  21. Circle(C, (0.15 * a)) → p
  22. Intersect(p, j, 1) → G
  23. Intersect(p, e, 1) → H
  24. Intersect(p, a, 1) → I
  25. Circle(A, (0.15 * a)) → q
  26. Circle(B, (0.15 * a)) → r
  27. Intersect(q, e, 1) → J
  28. Intersect(q, f, 1) → K
  29. CircleSector(C, H, I) → s
  30. Dilate(A, If[2 < g ≤ 3, 3 - g, 0], B) → A'
  31. Segment(A, A') → m
  32. Rotate(s, If[3.5 < g ≤ 4.5, ((-(g - 3.5)) * π / 2), -1.5707963267948966], C) → s'
  33. Dilate(A, If[5.5 < g ≤ 6.5, 6.5 - g, 0], B) → A'_1
  34. Vector(A, A'_1) → u
  35. Rotate(B, If[4.5 < g ≤ 5.5, ((-(g - 4.5)) * π / 2), -1.5707963267948966], C) → B'_1
  36. Rotate(B', If[4.5 < g ≤ 5.5, ((-(g - 4.5)) * π / 2), -1.5707963267948966], C) → B''
  37. Segment(B'_1, B'') → k'
  38. Dilate(A, If[7.5 < g ≤ 8.5, 8.5 - g, 0], C) → A'_2
  39. Vector(A, A'_2) → v
  40. Dilate(A, If[9.5 < g ≤ 10.5, 10.5 - g, 0], C) → A'_3
  41. Vector(A, A'_3) → w
  42. Line(B, C) → n
  43. Circle(B, 1) → t
  44. Line(C, A) → f_1
  45. Circle(A, 1) → d_1
  46. Intersect(d_1, f_1, 2) → L
  47. Intersect(t, n, 1) → M
  48. Ray(A, L) → g_1
  49. Ray(B, M) → h_1
  50. Rotate(L, 67.5°, A) → L'
  51. Angle(L, A, L') → β
  52. Rotate(A, 67.5°, B) → A'_4
  53. Angle(A, B, A'_4) → γ
  54. Ray(A, L') → i_1
  55. Ray(B, A'_4) → j_1
  56. Circle(B, (0.25 * k)) → e_1
  57. Circle(A, (0.25 * k)) → k_1
  58. Intersect(e_1, h_1, 1) → N
  59. Intersect(e_1, j_1) → O,
  60. Intersect(k_1, i_1) → P,
  61. Intersect(k_1, g_1) → Q,
  62. CircleSector(A, Q, P) → p_1
  63. CircleSector(B, O, N) → q_1
  64. Intersect(i_1, j_1) → R
  65. OrthogonalLine(R, h_1) → l_1
  66. OrthogonalLine(R, g_1) → m_1
  67. Intersect(m_1, g_1) → S
  68. Intersect(h_1, l_1) → T
  69. Angle(B, T, R) → δ
  70. Angle(R, S, A) → ε
  71. Dilate(A, If[5.5 < g ≤ 6.5, 6.5 - g, 0], R) → A'_5
  72. Dilate(B, If[5.5 < g ≤ 6.5, 6.5 - g, 0], R) → B'_2
  73. Segment(A, A'_5) → n_1
  74. Segment(B, B'_2) → r_1
  75. Dilate(R, If[6.5 < g ≤ 7.5, 7.5 - g, 0], T) → R'
  76. Dilate(R, If[6.5 < g ≤ 7.5, 7.5 - g, 0], S) → R'_1
  77. Segment(R, R') → s_1
  78. Segment(R, R'_1) → t_1
  79. Rotate(T, If[8.5 < g ≤ 10.5, ((g - 8.5) * π), (2 * π)], R) → T'
  80. Rotate(p_1, If[7.5 < g ≤ 8.5, (((g - 7.5) * 67.5) * π / 180), (67.5 * π / 180)], A) → c_2
  81. Rotate(q_1, If[7.5 < g ≤ 8.5, (((-(g - 7.5)) * 67.5) * π / 180), (-67.5 * π / 180)], B) → d_2
  82. CircleArc(R, T, T') → e_2
  83. Circle(R, T) → f_2
  84. Line(R, A) → g_2
  85. OrthogonalLine(C, g_2) → h_2
  86. Intersect(g_2, h_2) → V
  87. Angle(A, V, C) → ζ
  88. Dilate(C, If[10.5 < g ≤ 11.5, 11.5 - g, 0], V) → C'_2
  89. Dilate(A, If[10.5 < g ≤ 11.5, 11.5 - g, 0], V) → A'_6
  90. Segment(A, A'_6) → i_2
  91. Segment(C, C'_2) → j_2
  92. Midpoint(R, A) → U
  93. Dilate(A, If[11.5 < g ≤ 12.5, 12.5 - g, 0], U) → A'_7
  94. Segment(A, A'_7) → k_2
  95. Dilate(U, If[12.5 < g ≤ 13.5, 13.5 - g, 0], R) → U'
  96. Segment(U, U') → l_2
  97. Dilate(V, If[13.5 < g ≤ 14.5, 14.5 - g, 0], A) → V'
  98. Vector(V, V') → u_1
  99. Translate(C, u_1) → C'_3
  100. Translate(C'_2, u_1) → W
  101. Segment(C'_3, W) → m_2
  102. Rotate(C'_3, If[14.5 < g ≤ 15.5, ((-(g - 14.5)) * π / 2), ((-π)) / 2], A) → Z
  103. Rotate(W, If[14.5 < g ≤ 15.5, ((-(g - 14.5)) * π / 2), ((-π)) / 2], A) → W'
  104. Segment(Z, W') → n_2
  105. Rotate(Z, If[15.5 < g ≤ 16.5, ((-(g - 15.5)) * π), (-π)], U) → Z'
  106. Rotate(W', If[15.5 < g ≤ 16.5, ((-(g - 15.5)) * π), (-π)], U) → W''
  107. Segment(Z', W'') → p_2