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Special Points of a Triangle

Special Points of a Triangle
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2017-04-27 12:50
TriangleCircleCoordinate GeometryGeometric Construction
This example illustrates several special points of a triangle, including the centroid, circumcenter, incenter, and orthocenter. Using GeoGebra commands such as Polygon, LineBisector, and Intersect, the properties and positions of these points are demonstrated. Suitable for middle and high school students learning geometric construction and related concepts.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. LineBisector(c) → d
  3. LineBisector(b) → e
  4. LineBisector(a) → f
  5. Intersect(d, e) → O
  6. AngularBisector(A, C, B) → g
  7. AngularBisector(B, A, C) → h
  8. AngularBisector(C, B, A) → i
  9. Intersect(g, h) → I
  10. Midpoint(a) → F
  11. Midpoint(c) → G_1
  12. Midpoint(b) → H_1
  13. Line(G_1, C) → j
  14. Line(H_1, B) → k
  15. Line(A, F) → l
  16. Intersect(j, k) → G
  17. OrthogonalLine(A, a) → m
  18. OrthogonalLine(C, c) → n
  19. OrthogonalLine(B, b) → p
  20. Intersect(m, n) → H
  21. Circle(O, A) → r
  22. Distance(I, b) → distanceEb
  23. Circle(I, distanceEb) → s
  24. Intersect(f, a) → K
  25. Intersect(d, c) → L
  26. Intersect(e, b) → M
  27. Angle(O, K, B) → α
  28. Angle(O, M, C) → β
  29. Angle(O, L, A) → γ
  30. Line(A, C) → q
  31. Line(C, B) → t
  32. Line(B, A) → a_1
  33. Intersect(p, q) → N
  34. Intersect(n, a_1) → P
  35. Intersect(m, t) → Q
  36. Angle(H, N, C) → δ
  37. Angle(H, Q, B) → ε
  38. Angle(H, P, A) → ζ
  39. Angle(B, C, I) → η
  40. Angle(I, C, A) → θ
  41. Angle(C, A, I) → ι
  42. Angle(I, A, B) → κ
  43. Angle(A, B, I) → λ
  44. Angle(I, B, C) → μ
  45. Polygon(Q, N, P) → poly2, p_1, q_1, n_1
  46. Distance(H, p_1) → distanceJp1
  47. Intersect(l, a) → R
  48. Intersect(k, b) → S
  49. Intersect(j, c) → T
  50. Segment(C, R) → b_1
  51. Segment(R, B) → d_1
  52. Segment(B, T) → e_1
  53. Segment(T, A) → f_1
  54. Segment(A, S) → g_1
  55. Segment(S, C) → h_1
  56. Segment(A, G) → i_1
  57. Segment(G, F) → j_1
  58. Segment(H_1, G) → k_1
  59. Segment(G, B) → l_1
  60. Segment(G_1, G) → m_1
  61. Segment(G, C) → r_1
  62. OrthogonalLine(H, p_1) → c_1
  63. Intersect(c_1, p_1) → D
  64. Circle(H, D) → s_1
  65. Segment(H, O) → t_1

AI Drawing Prompt Reference

Draw a triangle ABC and find its circumcenter O, incenter I, centroid F, and orthocenter H. Use midpoints and angle bisectors to locate F, I, and H. Draw a circle with O and A, and another circle with I and the distance to a side. Use angle bisectors and perpendicular lines to find feet K, L, M, and angles α, β, γ. Finally, draw altitudes from H to find feet N, P, Q, and angles δ, ε, ζ.
The prompt above was compiled from the original drawing commands and is for reference only.