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Inscribed Angle Theorem Dynamic Demonstration

Inscribed Angle Theorem Dynamic Demonstration
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2018-03-01 09:04
CircleAngleTrigonometryGeometric ConstructionSlider Animation
In collections Circles (Theorems)
This example dynamically demonstrates the Inscribed Angle Theorem, visually showing the relationship between inscribed angles and their corresponding central angles through rotation and intersection changes. Core mathematical concepts include the Inscribed Angle Theorem, central angles, rotation, similar triangles, etc. Construction highlights involve GeoGebra commands/objects like Circle, Point, CircleArc, Angle, Rotate, Ray, Dilate, Segment, Intersect, CircleSector, Vector, Translate, and Line. Suitable for high school mathematics teaching to enhance understanding and application of the theorem.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. CircleArc(A, D, C) → d
  5. Point(d) → E
  6. Point(d) → F
  7. Point(d) → G
  8. Point(d) → H
  9. Point(d) → I
  10. CircleArc(A, C, D) → e
  11. Angle(C, A, D) → α
  12. Rotate(C, α / 2, A) → C'
  13. Angle(C, A, C') → β
  14. Ray(A, C') → h
  15. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C) → A'
  16. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D) → A'_1
  17. Segment(A, A') → f
  18. Segment(A, A'_1) → g
  19. Segment(A, B) → i
  20. Segment(A, C) → j
  21. Circle(A, (0.3 * i)) → k
  22. Intersect(k, j) → K,
  23. Dilate(E, If[1 < a ≤ 2, 2 - a, 0], C) → E'
  24. Dilate(E, If[1 < a ≤ 2, 2 - a, 0], D) → E'_1
  25. Dilate(F, If[1 < a ≤ 2, 2 - a, 0], D) → F'
  26. Dilate(G, If[1 < a ≤ 2, 2 - a, 0], D) → G'
  27. Dilate(H, If[1 < a ≤ 2, 2 - a, 0], D) → H'
  28. Dilate(I, If[1 < a ≤ 2, 2 - a, 0], D) → I'
  29. Dilate(F, If[1 < a ≤ 2, 2 - a, 0], C) → F'_1
  30. Dilate(G, If[1 < a ≤ 2, 2 - a, 0], C) → G'_1
  31. Dilate(H, If[1 < a ≤ 2, 2 - a, 0], C) → H'_1
  32. Dilate(I, If[1 < a ≤ 2, 2 - a, 0], C) → I'_1
  33. Segment(E, E') → l
  34. Segment(E, E'_1) → m
  35. Segment(F, F'_1) → n
  36. Segment(F, F') → p
  37. Segment(G, G'_1) → q
  38. Segment(G, G') → r
  39. Segment(H, H'_1) → s
  40. Segment(H, H') → t
  41. Segment(I, I'_1) → b
  42. Segment(I, I') → f_1
  43. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C') → A'_2
  44. Segment(A, A'_2) → i_1
  45. Rotate(K, If[1 < a ≤ 2, ((a - 1) * α), α], A) → K'
  46. CircleSector(A, K, K') → c_1
  47. CircleSector(A, D, C) → d_1
  48. CircleSector(A, C, D) → e_1
  49. Rotate(K, If[0 < a ≤ 1, (a * α / 2), α / 2], A) → K'_1
  50. CircleSector(A, K, K'_1) → k_1
  51. Ray(E, C) → j_1
  52. Ray(F, C) → l_1
  53. Ray(G, C) → m_1
  54. Ray(H, C) → n_1
  55. Ray(I, C) → p_1
  56. Circle(E, (0.3 * i)) → r_1
  57. Circle(F, (0.3 * i)) → s_1
  58. Circle(G, (0.3 * i)) → t_1
  59. Circle(H, (0.3 * i)) → c_2
  60. Circle(I, (0.3 * i)) → d_2
  61. Intersect(r_1, j_1) → L,
  62. Intersect(n, l_1) → M
  63. Intersect(s_1, l_1) → N,
  64. Intersect(t_1, m_1) → O,
  65. Intersect(c_2, n_1) → P,
  66. Intersect(d_2, p_1) → Q,
  67. Rotate(Q, If[1 < a ≤ 2, ((a - 1) * α / 2), α / 2], I) → Q'
  68. CircleSector(I, Q, Q') → f_2
  69. Rotate(k_1, If[2 < a ≤ 3, ((a - 2) * α / 2), α / 2], A) → g_2
  70. Dilate(I, If[3 < a ≤ 4, 4 - a, 0], F) → I'_2
  71. Vector(I, I'_2) → u
  72. Translate(f_2, u) → h_2
  73. Line(I, n) → a_1
  74. Point(a_1) → S
  75. Angle(Q, I, S) → γ
  76. Rotate(h_2, If[3 < a ≤ 4, ((-(a - 3)) * ((2 * π) - γ)), γ - (2 * π)], I'_2) → k_2
  77. Dilate(A, If[3.5 < a ≤ 4.5, 4.5 - a, 0], I) → A'_3
  78. Vector(A, A'_3) → v
  79. Translate(g_2, v) → p_2
  80. Rotate(K, If[3.5 < a ≤ 4.5, ((a - 3.5) * α / 2), α / 2], A) → K'_2
  81. CircleSector(A, K, K'_2) → q_2
  82. Line(I, i_1) → b_1
  83. Point(b_1) → T
  84. Angle(Q, I, T) → δ
  85. Rotate(p_2, If[3.5 < a ≤ 4.5, ((a - 3.5) * ((2 * π) - δ)), (2 * π) - δ], A'_3) → r_2
  86. Dilate(F, If[4.5 < a ≤ 5.5, 5.5 - a, 0], G) → F'_2
  87. Vector(F, F'_2) → w
  88. Translate(k_2, w) → s_2
  89. Line(G, n) → i_2
  90. Point(i_2) → U
  91. Angle(O, G, U) → ε
  92. Rotate(s_2, If[4.5 < a ≤ 5.5, ((-(a - 4.5)) * ε), (-ε)], F'_2) → t_2
  93. Rotate(L, If[3 < a ≤ 4, ((a - 3) * α / 2), α / 2], E) → L'
  94. CircleSector(E, L, L') → c_3
  95. Dilate(E, If[5 < a ≤ 6, 6 - a, 0], H) → E'_2
  96. Vector(E, E'_2) → u_1
  97. Translate(c_3, u_1) → d_3
  98. Line(H, l) → j_2
  99. Point(j_2) → V
  100. Angle(P, H, V) → ζ
  101. Rotate(d_3, If[5 < a ≤ 6, ((a - 5) * ((2 * π) - ζ)), (2 * π) - ζ], E'_2) → e_3
  102. Dilate(A, If[5.5 < a ≤ 6.5, 6.5 - a, 0], E) → A'_4
  103. Vector(A, A'_4) → v_1
  104. Translate(q_2, v_1) → f_3
  105. Line(E, f) → l_2
  106. Point(l_2) → W
  107. Angle(W, E, C) → η
  108. Rotate(f_3, If[5.5 < a ≤ 6.5, ((a - 5.5) * η), η], A'_4) → g_3
  109. Dilate(I, If[6 < a ≤ 7, 7 - a, 0], F) → I'_3
  110. Vector(I, I'_3) → w_1
  111. Translate(r_2, w_1) → g_1
  112. Line(F, b) → h_1
  113. Point(h_1) → J
  114. Angle(J, F, N) → θ
  115. Rotate(g_1, If[6 < a ≤ 7, ((-(a - 6)) * ((2 * π) - θ)), θ - (2 * π)], I'_3) → q_1
  116. Rotate(K, If[6.5 < a ≤ 7.5, ((a - 6.5) * β), β], A) → K'_3
  117. CircleSector(A, K, K'_3) → e_2
  118. Dilate(H, If[7 < a ≤ 8, 8 - a, 0], I) → H'_2
  119. Vector(H, H'_2) → u_2
  120. Translate(e_3, u_2) → h_3
  121. Line(I, s) → m_2
  122. Point(m_2) → R
  123. Angle(R, I, Q) → κ
  124. Rotate(h_3, If[8 < a ≤ 9, ((a - 8) * κ), κ], I) → k_3
  125. Dilate(A, If[8 < a ≤ 9, 9 - a, 0], H) → A'_5
  126. Vector(A, A'_5) → v_2
  127. Translate(e_2, v_2) → p_3
  128. Line(H, f) → n_2
  129. Point(n_2) → Z
  130. Angle(Z, H, P) → λ
  131. Rotate(p_3, If[8 < a ≤ 9, ((a - 8) * λ), λ], A'_5) → q_3