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Feuerbach Point Dynamic Demonstration

Feuerbach Point Dynamic Demonstration
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2018-12-29 05:10
TriangleCircleGeometric ConstructionLocus
This example dynamically illustrates the property of the Feuerbach point of any triangle. It involves core mathematical concepts like triangle, incircle, midpoint, angle bisector, circle intersections, etc. Construction highlights commands such as Segment, Rotate, Circle, Intersect, Polygon, Incircle, Midpoint from the summary. Suitable for high school and above students to understand the relationship between the Feuerbach point and the nine-point circle, and the dynamic changes in geometric constructions.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Polygon(A, B, C) → t1, c, a_1, b
  7. Segment(A, B) → h
  8. Segment(B, C) → i
  9. Segment(C, A) → j
  10. Incircle(A, B, C) → d
  11. Midpoint(j) → D
  12. Midpoint(h) → E
  13. Midpoint(i) → F
  14. OrthogonalLine(C, h) → k
  15. OrthogonalLine(B, j) → l
  16. OrthogonalLine(A, i) → m
  17. Intersect(k, l) → G
  18. LineBisector(h) → n
  19. LineBisector(i) → p
  20. LineBisector(j) → q
  21. Line(A, B) → r
  22. Line(B, C) → s
  23. Line(C, A) → t
  24. Intersect(m, s) → H
  25. Intersect(l, t) → I
  26. Intersect(k, r) → J
  27. Midpoint(G, B) → K
  28. Midpoint(G, C) → L
  29. Midpoint(G, A) → M
  30. Circle(L, K, M) → e
  31. Intersect(d, e) → N, O, ,
  32. Segment(N, D) → f_1
  33. Segment(N, F) → g_1
  34. Segment(N, E) → h_1
  35. Max({f_1, g_1, h_1}) → u
  36. Circle(N, g_1) → c_1
  37. Intersect(c_1, f_1, 1) → P
  38. Angle(F, N, D) → β
  39. Line(P, h_1) → i_1
  40. Rotate(N, If[0 < v ≤ 1, (v * β), β], N) → N'
  41. Rotate(F, If[0 < v ≤ 1, (v * β), β], N) → F'
  42. Segment(N', F') → j_1
  43. Dilate(N', If[1 < v ≤ 2, 2 - v, 0], F') → N''
  44. Vector(N, N'') → w
  45. Translate(N, w) → N'_1
  46. Translate(E, w) → E'
  47. Segment(N'_1, E') → k_1
  48. Line(F', h_1) → l_1
  49. Point(l_1) → Q
  50. Angle(D, F', Q) → γ
  51. Rotate(N'_1, If[2 < v ≤ 3, ((-(v - 2)) * γ), (-γ)], F') → R
  52. Rotate(E', If[2 < v ≤ 3, ((-(v - 2)) * γ), (-γ)], F') → E''
  53. Segment(R, E'') → m_1
  54. Segment(N, F) → n_1
  55. Segment(N, E) → p_1
  56. Segment(N, D) → q_1
  57. Circle(N, E) → d_1
  58. Intersect(d_1, n_1, 1) → S
  59. Angle(E, N, S) → δ
  60. Rotate(N, If[0 < v ≤ 1, (v * δ), δ], N) → N'_2
  61. Rotate(E, If[0 < v ≤ 1, (v * δ), δ], N) → E'_1
  62. Segment(N'_2, E'_1) → r_1
  63. Dilate(N, If[1 < v ≤ 2, 2 - v, 0], E'_1) → N'_3
  64. Vector(N, N'_3) → u_1
  65. Translate(N, u_1) → N'_4
  66. Translate(D, u_1) → D'
  67. Segment(N'_4, D') → s_1
  68. Angle(F, N, D) → ε
  69. Rotate(N'_4, If[2 < v ≤ 3, ((-(v - 2)) * ε), (-ε)], E'_1) → T
  70. Rotate(D', If[2 < v ≤ 3, ((-(v - 2)) * ε), (-ε)], E'_1) → D''
  71. Segment(T, D'') → t_1
  72. Segment(N, E) → b_1
  73. Segment(N, F) → e_1
  74. Segment(N, D) → f_2
  75. Angle(D, N, E) → ζ
  76. Rotate(N, If[0 < v ≤ 1, (v * ζ), ζ], N) → N'_5
  77. Rotate(D, If[0 < v ≤ 1, (v * ζ), ζ], N) → D'_1
  78. Segment(N'_5, D'_1) → g_2
  79. Dilate(N, If[1 < v ≤ 2, 2 - v, 0], D'_1) → N'_6
  80. Vector(N, N'_6) → v_1
  81. Translate(N, v_1) → N'_7
  82. Translate(F, v_1) → F'_1
  83. Segment(N'_7, F'_1) → h_2
  84. Angle(D'_1, N, F) → η
  85. Rotate(N'_7, If[2 < v ≤ 3, ((-(v - 2)) * η), (-η)], D'_1) → W
  86. Rotate(F'_1, If[2 < v ≤ 3, ((-(v - 2)) * η), (-η)], D'_1) → Z
  87. Segment(W, Z) → i_2