MathPlotAI Logo MathPlotAI

Dynamic Illustration of a Special Theorem on Concurrent Chords in a Circle

Dynamic Illustration of a Special Theorem on Concurrent Chords in a Circle
Loading the math board and drawing, please wait…
2016-12-03 09:24
CircleAngleGeometric Construction
In collections Circles (Theorems)
This example dynamically illustrates a special theorem concerning a set of three concurrent chords in a circle. Key mathematical concepts include circles, angles, and geometric construction. Using GeoGebra commands such as Circle, Point, Rotate, Intersect, etc., the properties of these chords can be observed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Radius(c) → o
  3. Point(c) → D
  4. Point(c) → C
  5. Segment(C, D) → f
  6. Point(f) → E
  7. Rotate(C, π / 3, E) → C'
  8. Line(E, C') → g
  9. Intersect(c, g, 2) → F
  10. Line(F, g_1) → l_1
  11. Point(l_1) → G_1
  12. Angle(C, E, C') → α
  13. Rotate(C', π / 3, E) → C''
  14. Angle(C', E, C'') → β
  15. Line(E, C'') → h
  16. Segment(C, E) → i
  17. Segment(E, D) → j
  18. Intersect(c, g, 1) → G
  19. Segment(E, F) → k
  20. Segment(E, G) → l
  21. Intersect(c, h, 1) → H
  22. Intersect(c, h, 2) → I
  23. Segment(E, H) → m
  24. Segment(E, I) → n
  25. Min(list) → a
  26. Circle(E, (0.28 * a)) → d
  27. Intersect(d, k, 1) → J
  28. Intersect(f, i) → K
  29. Intersect(f, i) → L
  30. Intersect(f, i) → M
  31. Intersect(f, i) → N
  32. Intersect(d, i) → O,
  33. Intersect(d, m) → P, H_1
  34. CircleSector(E, P, O) → e
  35. Segment(D, E) → p
  36. Segment(E, C) → q
  37. Dilate(I, If[0 < b ≤ 1, 1 - b, 0], E) → I'
  38. Segment(I, I') → r
  39. Intersect(c, r, 1) → R
  40. Dilate(E, If[1 < b ≤ 2, 2 - b, 0], H) → E'
  41. Segment(E, E') → s
  42. Dilate(F, If[2 < b ≤ 3, 3 - b, 0], E) → F'
  43. Segment(F, F') → t
  44. Dilate(E, If[3 < b ≤ 4, 4 - b, 0], G) → E'_1
  45. Segment(E, E'_1) → f_1
  46. Rotate(e, If[5 < b ≤ 6, ((b - 5) * π / 3), π / 3], E) → e'
  47. Rotate(e', If[7 < b ≤ 8, ((b - 7) * π / 3), π / 3], E) → e''
  48. Circle(Q, p) → c_1
  49. Intersect(c_1, g_1, 2) → T
  50. Circle(T, t) → d_1
  51. Intersect(d_1, g_1, 2) → U
  52. Circle(U, s) → e_1
  53. Intersect(e_1, g_1, 2) → V
  54. Circle(S, r) → k_1
  55. Line(E, g_1) → i_1
  56. Line(D, g_1) → j_1
  57. Dilate(S, If[9 < b ≤ 10, 10 - b, 0], Q) → S'
  58. Segment(S, S') → m_1
  59. Dilate(Q, If[10 < b ≤ 11, 11 - b, 0], V) → Q'
  60. Vector(Q, Q') → w
  61. Translate(S, w) → S'_1
  62. Translate(S', w) → S''
  63. Segment(S'_1, S'') → n_1
  64. Point(j_1) → B_1
  65. Point(j_1) → C_1
  66. Angle(B_1, D, E) → γ
  67. Dilate(D, If[11 < b ≤ 12, 12 - b, 0], Q) → D'
  68. Vector(D, D') → u_1
  69. Translate(D, u_1) → D'_1
  70. Translate(E, u_1) → E'_2
  71. Segment(D'_1, E'_2) → p'
  72. Rotate(D'_1, If[11 < b ≤ 12, ((-(b - 11)) * γ), (-γ)], D'_1) → D_1
  73. Rotate(E'_2, If[11 < b ≤ 12, ((-(b - 11)) * γ), (-γ)], D'_1) → E_1
  74. Segment(D_1, E_1) → p''
  75. Dilate(F, If[12 < b ≤ 13, 13 - b, 0], T) → F'_1
  76. Vector(F, F'_1) → v_1
  77. Translate(F, v_1) → F'_2
  78. Translate(F', v_1) → F''
  79. Angle(E, F, G_1) → δ
  80. Segment(F'_2, F'') → t'
  81. Rotate(F'_2, If[12 < b ≤ 13, ((b - 12) * δ), δ], F'_1) → F_1
  82. Rotate(F'', If[12 < b ≤ 13, ((b - 12) * δ), δ], F'_1) → F'''
  83. Segment(F_1, F''') → t''
  84. Line(H, j_1) → r_1
  85. Point(r_1) → I_1
  86. Dilate(H, If[13 < b ≤ 14, 14 - b, 0], U) → H'
  87. Angle(I_1, H, E) → ε
  88. Vector(H, H') → w_1
  89. Translate(E, w_1) → E'_3
  90. Translate(E', w_1) → E''
  91. Segment(E'_3, E'') → s'
  92. Rotate(E'_3, If[13 < b ≤ 14, ((-(b - 13)) * ε), (-ε)], E'') → J_1
  93. Rotate(E'', If[13 < b ≤ 14, ((-(b - 13)) * ε), (-ε)], E'') → E'''
  94. Segment(J_1, E''') → s''
  95. Dilate(G, If[14 < b ≤ 15, 15 - b, 0], S) → G'
  96. Vector(G, G') → a_1
  97. Translate(E, a_1) → E'_4
  98. Translate(E'_1, a_1) → K_1
  99. Segment(E'_4, K_1) → s_1
  100. Line(G, j_1) → t_1
  101. Point(t_1) → L_1
  102. Angle(L_1, G, E) → ζ
  103. Rotate(E'_4, If[14 < b ≤ 15, ((-(b - 14)) * ζ), (-ζ)], K_1) → M_1
  104. Rotate(K_1, If[14 < b ≤ 15, ((-(b - 14)) * ζ), (-ζ)], K_1) → N_1
  105. Segment(M_1, N_1) → b_1
  106. Circle(S, b_1) → c_2
  107. Intersect(c_2, h_1, 2) → O_1
  108. Circle(O_1, r) → p_1
  109. Intersect(p_1, h_1, 2) → W
  110. Dilate(R, If[15 < b ≤ 16, 16 - b, 0], O_1) → R'
  111. Vector(I, R') → q_1
  112. Translate(I, q_1) → I'_1
  113. Translate(I', q_1) → I''
  114. Segment(I'_1, I'') → r'
  115. Line(I, i_1) → f_2
  116. Point(f_2) → Z
  117. Angle(E, I, Z) → η
  118. Rotate(I'_1, If[15 < b ≤ 16, ((b - 15) * η), η], I'_1) → A_1
  119. Rotate(I'', If[15 < b ≤ 16, ((b - 15) * η), η], I'_1) → I'''
  120. Segment(A_1, I''') → r''
  121. Dilate(E, If[16 < b ≤ 17, 17 - b, 0], W) → E'_5
  122. Vector(E, E'_5) → u_2
  123. Translate(E, u_2) → E'_6
  124. Translate(C, u_2) → C'_1
  125. Segment(E'_6, C'_1) → q'
  126. Point(i_1) → P_1
  127. Angle(P_1, E, C) → θ
  128. Rotate(E'_6, If[16 < b ≤ 17, ((-(b - 16)) * θ), (-θ)], E'_6) → Q_1
  129. Rotate(C'_1, If[16 < b ≤ 17, ((-(b - 16)) * θ), (-θ)], E'_6) → R_1
  130. Segment(Q_1, R_1) → q''
  131. Dilate(G, If[17 < b ≤ 18, 18 - b, 0], E) → G'_1
  132. Dilate(D, If[17 < b ≤ 18, 18 - b, 0], E) → D'_2
  133. Dilate(R, If[17 < b ≤ 18, 18 - b, 0], E) → R'_1
  134. Dilate(F, If[17 < b ≤ 18, 18 - b, 0], E) → F'_3
  135. Dilate(C, If[17 < b ≤ 18, 18 - b, 0], E) → C'_2
  136. Dilate(H, If[17 < b ≤ 18, 18 - b, 0], E) → H'_1
  137. Segment(D, D'_2) → g_2
  138. Segment(I, R'_1) → h_2
  139. Segment(F, F'_3) → i_2
  140. Segment(C, C'_2) → j_2
  141. Segment(H, H'_1) → k_2
  142. Segment(G, G'_1) → l_2