MathPlotAI Logo MathPlotAI

Animation 248: Composite Construction of Triangle Incircle, Circumcircle, Rotati

Animation 248: Composite Construction of Triangle Incircle, Circumcircle, Rotati
Loading the math board and drawing, please wait…
2017-04-23 23:26
TriangleCircleTransformationGeometric ConstructionSlider Animation
① This example demonstrates a dynamic geometric animation involving a triangle, its circumcircle, incircle, and composite transformations (rotation and dilation), with sliders controlling parameters to observe how the figure changes with angles or scale factors. ② Core mathematical concepts include the incenter, circumcenter, incircle, circumcircle, similarity and homothety (dilation), rotational symmetry, angle relationships, right triangle properties, and the integration of scatter plots with arithmetic operations, reflecting the fusion of geometry and algebra. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Line, Slope, PointIn, Angle, Segment, Min, Circle, Intersect, Incircle, Center, Polygon, Rotate, Ray, CircleSector, Dilate, OrthogonalLine, CircleArc, and LineBisector are used, creating objects A, B, d, m, b, a, c, C, e, f, α, β, γ, g, h, i, list1, j, q, p, k, D, E, F, G, H, I_2, J, K, L, N, O, P, δ, ε, η, θ, r, inc, and I. Sliders drive rotation and scaling, updating the figure dynamically. ④ Suitable for high school to early college geometry instruction, for exploring triangle centers, circle properties, and composite transformations, and as a dynamic geometry courseware or mathematical experiment material.

GeoGebra Commands & Parameters

  1. Line(A, B) → d
  2. Slope(d) → a
  3. PointIn(c) → C
  4. Line(A, C) → e
  5. Line(B, C) → f
  6. Angle(B, A, C) → α
  7. Angle(A, C, B) → β
  8. Angle(C, B, A) → γ
  9. Segment(A, C) → g
  10. Segment(C, B) → h
  11. Segment(B, A) → i
  12. Min(list1) → j
  13. Circle(C, (0.2 * j)) → q
  14. Circle(A, (0.2 * j)) → p
  15. Circle(B, (0.2 * j)) → k
  16. Intersect(q, f, 2) → D
  17. Intersect(q, e, 1) → E
  18. Intersect(q, e, 2) → F
  19. Intersect(p, e, 2) → G
  20. Intersect(p, d, 1) → H
  21. Intersect(k, f, 2) → I_2
  22. Intersect(k, d, 2) → J
  23. Intersect(d, i) → K
  24. Intersect(d, i) → L
  25. Intersect(k, f, 1) → N
  26. Intersect(p, e, 1) → O
  27. Intersect(p, d, 2) → P
  28. Angle(D, C, E) → δ
  29. Angle(O, A, P) → ε
  30. Angle(J, B, I_2) → η
  31. Angle(G, A, H) → θ
  32. Incircle(A, B, C) → r
  33. Center(r) → I
  34. Line(C, I) → l
  35. Line(A, I) → n
  36. Line(B, I) → s
  37. Intersect(e, s) → R
  38. Intersect(d, l) → S
  39. Intersect(f, n) → T
  40. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  41. Rotate(D, δ / 2, C) → D'
  42. Angle(D, C, D') → κ
  43. Rotate(J, η / 2, B) → J'
  44. Angle(J, B, J') → λ
  45. Ray(C, D') → t
  46. Ray(B, J') → e_1
  47. Rotate(O, ε / 2, A) → O'
  48. Angle(O, A, O') → ι
  49. Ray(A, O') → d_1
  50. Segment(C, S) → f_1
  51. Segment(A, T) → g_1
  52. Segment(B, R) → h_1
  53. Intersect(q, f_1, 1) → U
  54. Intersect(q, f, 1) → V
  55. Intersect(p, g_1, 1) → W
  56. Intersect(k, h_1, 1) → Z
  57. CircleSector(A, P, W) → q_1
  58. CircleSector(B, I_2, Z) → p_1
  59. Dilate(C, If[0 < m ≤ 1, 1 - m, 0], S) → C'
  60. Dilate(B, If[2 < m ≤ 3, 3 - m, 0], R) → B'
  61. Segment(B, B') → j_1
  62. Rotate(p_1, If[3 < m ≤ 4, ((m - 3) * γ / 2), γ / 2], B) → s_1
  63. Dilate(A, If[4 < m ≤ 5, 5 - m, 0], T) → A'
  64. Segment(A, A') → l_1
  65. Rotate(q_1, If[5 < m ≤ 6, ((m - 5) * α / 2), α / 2], A) → t_1
  66. OrthogonalLine(I, f) → m_1
  67. OrthogonalLine(I, d) → n_1
  68. OrthogonalLine(I, e) → a_2
  69. Intersect(f, m_1) → M
  70. Intersect(d, n_1) → A_1
  71. Intersect(e, a_2) → B_1
  72. Segment(I, M) → b_2
  73. Circle(M, (0.2 * b_2)) → c_2
  74. Intersect(c_2, f, 1) → C_1
  75. Intersect(c_2, b_2, 1) → D_1
  76. Intersect(c_2, f, 2) → E_1
  77. CircleSector(M, E_1, D_1) → d_2
  78. Dilate(I, If[6 < m ≤ 7, 7 - m, 0], M) → Q'
  79. Segment(I, Q') → e_2
  80. Rotate(d_2, If[7.5 < m ≤ 8.5, ((m - 7.5) * π / 2), 1.5707963267948966], M) → f_2
  81. Segment(I, B_1) → g_2
  82. Segment(I, A_1) → h_2
  83. Circle(B_1, (0.2 * g_2)) → k_2
  84. Circle(A_1, (0.2 * h_2)) → p_2
  85. Intersect(k_2, e, 2) → F_1
  86. Intersect(k_2, g_2, 1) → G_1
  87. Intersect(k_2, e, 1) → H_1
  88. Intersect(p_2, d, 2) → I_1
  89. Intersect(p_2, h_2, 1) → J_1
  90. Intersect(p_2, d, 1) → K_1
  91. Dilate(I, If[8.5 < m ≤ 9.5, 9.5 - m, 0], B_1) → Q'_1
  92. Segment(I, Q'_1) → i_2
  93. CircleSector(B_1, H_1, G_1) → q_2
  94. CircleSector(A_1, I_1, J_1) → r_2
  95. Rotate(q_2, If[10 < m ≤ 11, ((m - 10) * π / 2), 1.5707963267948966], B_1) → s_2
  96. Dilate(I, If[11 < m ≤ 12, 12 - m, 0], A_1) → Q'_2
  97. Segment(I, Q'_2) → j_2
  98. Rotate(r_2, If[12.5 < m ≤ 13.5, ((m - 12.5) * π / 2), 1.5708], A_1) → t_2
  99. Angle(M, I, B_1) → ζ
  100. Angle(B_1, I, A_1) → μ
  101. Angle(A_1, I, M) → ν
  102. Rotate(I, If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q'_3
  103. Rotate(Q', If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q''
  104. Segment(Q'_3, Q'') → l_2
  105. Rotate(Q'_3, If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → L_1
  106. Rotate(Q'', If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → Q'''
  107. Segment(L_1, Q''') → m_2
  108. Rotate(A_1, If[16 < m ≤ 18, ((m - 16) * π), 6.28319], I) → M'
  109. Intersect(m_1, f) → Q
  110. Circle(I, Q) → d_3
  111. Line(A, B) → n_2
  112. CircleArc(I, A_1, M') → c_3
  113. Circle(A, I, B) → k_1
  114. LineBisector(I, B) → i_1
  115. LineBisector(I, A) → r_1
  116. Intersect(i_1, r_1) → M_1
  117. Dilate(M_1, If[19 < m ≤ 20, 20 - m, 0], B) → N_1
  118. Segment(M_1, N_1) → f_3
  119. Rotate(B, If[20 < m ≤ 22, ((m - 20) * π), (2 * π)], M_1) → B'_1
  120. CircleArc(M_1, B, B'_1) → e_3
  121. Dilate(C, If[22 < m ≤ 23, 23 - m, 0], I) → C'_1
  122. Segment(C, C'_1) → g_3
  123. Dilate(I, If[23 < m ≤ 24, 24 - m, 0], M_1) → I'
  124. Segment(I, I') → h_3

AI Drawing Prompt Reference

Create a dynamic geometric animation showing a triangle, its circumcircle, and incircle, with sliders controlling rotation and scaling parameters to observe how the figure changes with angles or scale factors. The animation includes triangle centers, circle properties, similarity and homothety (dilation), rotational symmetry, angle relationships, right triangle properties, and the integration of geometry and algebra, such as scatter plots with arithmetic operations.
The prompt above was compiled from the original drawing commands and is for reference only.