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Dynamic Illustration of Triangle Incircle and Midpoints

Dynamic Illustration of Triangle Incircle and Midpoints
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2017-06-21 10:12
TriangleCircleGeometric ConstructionSlider AnimationLocus
This example dynamically illustrates the geometric relationship among a triangle's incircle and the midpoints of two of its sides. Key math concepts include incircle, similar triangles, and geometric transformations. Construction highlights the use of GeoGebra commands like Midpoint, Segment, Dilate, Rotate, Polygon, Angle, Ray, Intersect, OrthogonalLine, Min, Circle, CircleSector, CircleArc, and Line, along with objects A, B, C, etc. Suitable for high school mathematics education or exploration by geometry enthusiasts.

GeoGebra Commands & Parameters

  1. Midpoint(A, B) → D
  2. Midpoint(B, C) → E
  3. Midpoint(C, A) → F
  4. Segment(A, B) → a
  5. Segment(B, C) → b
  6. Segment(C, A) → c
  7. Segment(C, D) → d
  8. Segment(A, E) → e
  9. Segment(B, F) → f
  10. Dilate(A, If[0 < g ≤ 1, 1 - g, 0], D) → A'
  11. Segment(A, A') → h
  12. Rotate(A, If[1 < g ≤ 2, ((-(g - 1)) * π), (-π)], D) → A'_1
  13. Rotate(A', If[1 < g ≤ 2, ((g - 1) * π), 3.141592653589793], D) → A''
  14. Segment(A'_1, A'') → h'
  15. Dilate(B, If[2 < g ≤ 3, 3 - g, 0], E) → B'
  16. Segment(B, B') → i
  17. Rotate(B, If[3 < g ≤ 4, ((-(g - 3)) * π), (-π)], E) → B'_1
  18. Rotate(B', If[3 < g ≤ 4, ((g - 3) * π), 3.141592653589793], E) → B''
  19. Segment(B'_1, B'') → i'
  20. Dilate(C, If[4 < g ≤ 5, 5 - g, 0], F) → C'
  21. Segment(C, C') → j
  22. Rotate(C, If[5 < g ≤ 6, ((-(g - 5)) * π), (-π)], F) → C'_1
  23. Rotate(C', If[5 < g ≤ 6, ((g - 5) * π), 3.141592653589793], F) → C''
  24. Segment(C'_1, C'') → j'
  25. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  26. Angle(B, A, C) → α
  27. Angle(A, C, B) → β
  28. Angle(C, B, A) → γ
  29. Rotate(B, α / 2, A) → B'_2
  30. Angle(B, A, B'_2) → δ
  31. Ray(A, B'_2) → k
  32. Rotate(C, γ / 2, B) → C'_3
  33. Angle(C, B, C'_3) → ε
  34. Ray(B, C'_3) → l
  35. Intersect(k, l) → O
  36. Segment(C, B) → m
  37. Segment(C, A) → n
  38. Intersect(m, k) → P
  39. Intersect(l, n) → Q
  40. Segment(A, B) → s
  41. OrthogonalLine(O, s) → t
  42. OrthogonalLine(O, m) → f_1
  43. Segment(C, A) → g_1
  44. OrthogonalLine(O, g_1) → h_1
  45. Intersect(h_1, g_1) → R
  46. Intersect(t, s) → S
  47. Intersect(f_1, m) → T
  48. Dilate(O, If[10 < g ≤ 11, 11 - g, 0], S) → O'
  49. Segment(O, O') → i_1
  50. Min({g_1, s, m}) → o
  51. Circle(A, (0.25 * o)) → d_1
  52. Circle(B, (0.25 * o)) → e_1
  53. Intersect(d_1, g_1) → U,
  54. Intersect(d_1, s) → V,
  55. Intersect(e_1, m) → W,
  56. Intersect(e_1, s) → Z,
  57. Intersect(d_1, k) → A_1,
  58. Intersect(e_1, l) → B_1,
  59. CircleSector(A, V, A_1) → k_1
  60. CircleSector(B, W, B_1) → p_1
  61. Dilate(A, If[6 < g ≤ 7, 7 - g, 0], O) → A'_3
  62. Segment(A, A'_3) → j_1
  63. Dilate(B, If[7 < g ≤ 8, 8 - g, 0], O) → B'_3
  64. Segment(B, B'_3) → p
  65. Rotate(k_1, If[8 < g ≤ 9, ((g - 8) * δ), δ], A) → q
  66. Rotate(p_1, If[9 < g ≤ 10, ((g - 9) * ε), ε], B) → r
  67. Angle(O, S, A) → ζ
  68. Rotate(S, If[11 < g ≤ 13, ((g - 11) * π), (2 * π)], O) → S'
  69. CircleArc(O, S, S') → q_1
  70. Circle(O, S) → r_1
  71. Segment(F, E) → l_1
  72. Line(C, S) → m_1
  73. Intersect(m_1, l_1) → G
  74. Dilate(C, If[13 < g ≤ 14, 14 - g, 0], S) → C'_2
  75. Segment(C, C'_2) → n_1
  76. Dilate(E, If[14 < g ≤ 15, 15 - g, 0], G) → E'
  77. Segment(E, E') → s_1
  78. Dilate(G, If[15 < g ≤ 16, 16 - g, 0], F) → G'
  79. Segment(G, G') → t_1
  80. Dilate(D, If[16 < g ≤ 17, 17 - g, 0], O) → D'
  81. Segment(D, D') → f_2
  82. Dilate(O, If[17 < g ≤ 18, 18 - g, 0], G) → O'_1
  83. Segment(O, O'_1) → g_2