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The Euler Line & Centers of the Triangle

The Euler Line & Centers of the Triangle
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2017-01-27 22:33
TriangleCircleVectorCoordinate GeometryGeometric Construction
This example demonstrates the Euler line of a triangle and its five significant centers: centroid, circumcenter, orthocenter, incenter, and nine-point center. Core mathematical concepts include centroid, circumcenter, orthocenter, incenter, nine-point circle, Euler line, vectors, and coordinate geometry. Construction highlights involve Polygon for triangles, Circle for circumcircle and nine-point circle, Intersect for intersections, Midpoint for midpoints, Line for lines, OrthogonalLine for perpendiculars, and Segment for line segments. Suitable for high school and above students to learn triangle geometry properties, competitive mathematics, and geometric constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Circle(C, D) → d
  3. Circle(B, E) → e
  4. Circle(A, F) → f
  5. Intersect(d, b, 1) → G
  6. Intersect(d, a, 1) → H
  7. Intersect(e, a, 1) → I
  8. Intersect(e, c, 1) → J
  9. Intersect(f, c, 1) → K
  10. Intersect(f, b, 1) → L
  11. Midpoint(L, K) → M
  12. Midpoint(J, I) → N
  13. Midpoint(H, G) → O
  14. Line(C, O) → g
  15. Line(B, N) → h
  16. Line(A, M) → i
  17. Intersect(g, h) → lncenter
  18. Intersect(g, i) → Incenter_.
  19. Intersect(h, i) → Incenter
  20. OrthogonalLine(Incenter, c) → j
  21. OrthogonalLine(Incenter, a) → k
  22. OrthogonalLine(Incenter, b) → l
  23. Intersect(j, c) → S
  24. Circle(Incenter, S) → p
  25. OrthogonalLine(Incenter_., c) → m
  26. OrthogonalLine(Incenter_., a) → n
  27. OrthogonalLine(Incenter_., b) → q
  28. Intersect(m, c) → T
  29. Circle(Incenter_., T) → r
  30. OrthogonalLine(lncenter, c) → s
  31. OrthogonalLine(lncenter, b) → t
  32. OrthogonalLine(lncenter, a) → f_1
  33. Intersect(s, c) → U
  34. Circle(lncenter, U) → c_1
  35. Midpoint(b) → Pm_3
  36. Midpoint(a) → Pm_2
  37. Midpoint(c) → Pm_1
  38. Line(C, Pm_1) → g_1
  39. Line(Pm_2, A) → h_1
  40. Line(Pm_3, B) → i_1
  41. Intersect(g_1, h_1) → Centroid
  42. OrthogonalLine(C, c) → j_1
  43. OrthogonalLine(A, a) → k_1
  44. OrthogonalLine(B, b) → l_1
  45. Intersect(j_1, k_1) → Orthocenter
  46. Intersect(l_1, b) → Ph_3
  47. Intersect(j_1, c) → Ph_1
  48. Intersect(k_1, a) → Ph_2
  49. Segment(Orthocenter, C) → m_1
  50. Segment(Orthocenter, B) → n_1
  51. Segment(Orthocenter, A) → p_1
  52. Midpoint(n_1) → P_H2
  53. Midpoint(p_1) → P_H1
  54. Midpoint(m_1) → P_H3
  55. OrthogonalLine(Pm_2, a) → q_1
  56. OrthogonalLine(Pm_1, c) → r_1
  57. OrthogonalLine(Pm_3, b) → s_1
  58. Intersect(q_1, r_1) → Circumcenter
  59. Circle(Circumcenter, A) → Cirumcircle
  60. Midpoint(Orthocenter, Circumcenter) → NinePointCenter
  61. Circle(NinePointCenter, Pm_1) → NinePointCircle
  62. Line(Circumcenter, Orthocenter) → t_1
  63. Line(Circumcenter, Orthocenter) → TheEulerLine