MathPlotAI Logo MathPlotAI

Circumcircle and Quadrilateral Animation

Circumcircle and Quadrilateral Animation
Loading the math board and drawing, please wait…
2017-04-15 10:51
CirclePolygonTrigonometryGeometric Construction
This example demonstrates the construction and properties of the circumcircle of a quadrilateral through animation. It involves core concepts like circumcircle, inscribed angle theorem, cosine rule, and quadrilateral classification. GeoGebra commands used include Circle, Point, Line, Intersect, CircleArc, Segment, Dilate, Rotate, Ray, Min, CircleSector, OrthogonalLine, Angle, Vector, Translate, Mirror, Polygon. Suitable for high school geometry education to understand the geometric relationship between quadrilaterals and circles.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Line(C, A) → a
  4. Intersect(c, a, 2) → D
  5. CircleArc(A, C, D) → d
  6. CircleArc(A, D, C) → e
  7. Point(e) → E
  8. Segment(C, D) → h
  9. Dilate(C, If[0 < b ≤ 1, 1 - b, 0], A) → C'
  10. Segment(C, C') → i
  11. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], D) → A'
  12. Segment(A, A') → j
  13. Rotate(C, If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C'_1
  14. Rotate(C', If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C''
  15. Segment(C'_1, C'') → i'
  16. Rotate(C, If[3 < b ≤ 4, ((b - 3) * π), π], A) → C'_2
  17. CircleArc(A, C, C'_2) → k
  18. Dilate(D, If[5 < b ≤ 6, 6 - b, 0], E) → D'
  19. Segment(D, D') → f
  20. Ray(D, E) → g
  21. Point(g) → F
  22. Ray(E, F) → l
  23. Dilate(C, If[5 < b ≤ 6, 6 - b, 0], E) → C'_3
  24. Segment(C, C'_3) → m
  25. Min({m, f}) → n
  26. Circle(E, (0.2 * n)) → p
  27. Intersect(p, l) → G,
  28. Intersect(c, m, 1) → H
  29. Intersect(p, m) → I,
  30. CircleSector(E, G, I) → q
  31. Rotate(q, If[6.5 < b ≤ 7.5, ((b - 6.5) * π / 2), π / 2], E) → q'
  32. Point(cc) → J
  33. Line(C, D) → r
  34. OrthogonalLine(J, r) → s
  35. Intersect(s, r) → K
  36. Line(C, E) → t
  37. OrthogonalLine(J, t) → f_1
  38. Intersect(t, f_1) → L
  39. Line(K, L) → g_1
  40. Line(J, E) → h_1
  41. Intersect(g_1, h_1) → M
  42. Angle(A, M, J) → α
  43. Line(A, M) → i_1
  44. Circle(C, 1) → c_1
  45. Circle(E, 1) → d_1
  46. Intersect(d_1, t, 2) → N
  47. Intersect(c_1, t, 1) → O
  48. Ray(E, N) → j_1
  49. Ray(C, O) → k_1
  50. Dilate(J, If[8 < b ≤ 9, 9 - b, 0], K) → J'
  51. Segment(J, J') → l_1
  52. Dilate(E, If[9 < b ≤ 10, 10 - b, 0], K) → E'
  53. Vector(E, E') → u
  54. Translate(q', u) → q''
  55. Line(E, r) → m_1
  56. Point(m_1) → P
  57. Angle(D, E, P) → β
  58. Rotate(q'', If[10 < b ≤ 11, ((b - 10) * β), β], K) → q'''
  59. Mirror(J, r) → J'_1
  60. Dilate(K, If[9 < b ≤ 10, 10 - b, 0], J'_1) → K'
  61. Segment(K, K') → n_1
  62. Dilate(J, If[11 < b ≤ 12, 12 - b, 0], L) → J'_2
  63. Segment(J, J'_2) → p_1
  64. Dilate(E, If[12 < b ≤ 13, 13 - b, 0], L) → E'_1
  65. Vector(E, E'_1) → v
  66. Translate(q', v) → q''_1
  67. Dilate(J, If[13 < b ≤ 14, 14 - b, 0], E) → J'_3
  68. Segment(J, J'_3) → q_1
  69. Dilate(K, If[14 < b ≤ 15, 15 - b, 0], M) → K'_1
  70. Segment(K, K'_1) → r_1
  71. Dilate(M, If[15 < b ≤ 16, 16 - b, 0], L) → M'
  72. Segment(M, M') → s_1
  73. Circle(A, 1) → e_1
  74. Circle(M, 1) → t_1
  75. Intersect(t_1, c, 1) → Q
  76. Intersect(t_1, i_1) → R, S
  77. Intersect(e_1, i_1) → T, U
  78. Dilate(A, If[16 < b ≤ 17, 17 - b, 0], M) → A'_1
  79. Segment(A, A'_1) → a_1
  80. Ray(M, S) → b_1
  81. Ray(A, T) → f_2
  82. Dilate(E, If[17 < b ≤ 18, 18 - b, 0], M) → E'_2
  83. Vector(E, E'_2) → w
  84. Translate(q', w) → q''_2
  85. Line(E, f_2) → g_2
  86. Point(g_2) → V
  87. Angle(V, E, L) → δ
  88. Rotate(q''_2, If[18 < b ≤ 19, ((-(b - 18)) * δ), (-δ)], M) → c_2
  89. Polygon(C, D, E) → poly1, e_2, c_3, d_2
  90. CircleArc(A, E, C) → h_2
  91. CircleArc(A, D, E) → k_2
  92. CircleArc(A, C, D) → p_2
  93. Polygon(C, D, E) → poly2, e_3, c_4, d_3
  94. Circle(C, 1) → q_2
  95. Circle(D, 1) → r_2
  96. Intersect(r_2, a) → A_1, B_1
  97. Ray(D, B_1) → i_2

AI Drawing Prompt Reference

Draw a quadrilateral and demonstrate the dynamic construction of its circumcircle. Start by drawing one side of the quadrilateral and its perpendicular bisector to find the circumcenter, then draw the circumcircle. Next, show how changing the shape of the quadrilateral affects the circumcircle by rotating and scaling its vertices. Finally, add some helper lines and angle markings to illustrate the inscribed angle theorem and cosine rule.
The prompt above was compiled from the original drawing commands and is for reference only.