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Longchamps Point: Geometric Properties and Construction in Triangle

Longchamps Point: Geometric Properties and Construction in Triangle
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2026-07-11 19:56
TriangleCirclePolygonGeometric Construction
In collections Besondere Punkte/Special Points
This GeoGebra worksheet illustrates the Longchamps point of a triangle and explores several of its curious properties. Using vertices A, B, C, the Polygon command creates triangle ABC; TriangleCenter computes the centroid, orthocenter, circumcenter and other triangle centers; Circle draws the circumcircle; Ellipse draws a related conic; Line and OrthogonalLine construct the Euler line and perpendiculars; Intersect finds intersection points; Segment draws the sides. The applet is ideal for high‑school or undergraduate geometry courses, math contest preparation, and teachers demonstrating triangle centers and Euler line relationships.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → T_{ABC}, c, a, b
  2. Ellipse(C, B, A) → Ωa
  3. Ellipse(A, C, B) → Ωb
  4. Ellipse(A, B, C) → Ωc
  5. Intersect(Ωb, Ωc, 2) → D
  6. Intersect(Ωb, Ωc, 1) → E
  7. Segment(D, E) → g
  8. Intersect(Ωa, Ωb, 2) → F
  9. Intersect(Ωa, Ωb, 1) → G_1
  10. Segment(F, G_1) → h
  11. Intersect(Ωa, Ωc, 1) → H_1
  12. Intersect(Ωa, Ωc, 2) → I_1
  13. Segment(H_1, I_1) → i
  14. TriangleCenter(A, B, C, 20) → Lo
  15. Line(A, a) → j
  16. Line(B, b) → k
  17. Line(C, c) → l
  18. Intersect(k, l) → A'
  19. Intersect(j, l) → B'
  20. Intersect(j, k) → C'
  21. OrthogonalLine(A', j) → m
  22. OrthogonalLine(C', l) → n
  23. OrthogonalLine(B', k) → p
  24. Circle(B, A, C) → q
  25. TriangleCenter(A, B, C, 1) → I
  26. TriangleCenter(A, B, C, 2) → G
  27. TriangleCenter(A, B, C, 4) → H
  28. TriangleCenter(A, B, C, 7) → Ge
  29. Line(Ge, I) → r
  30. Line(H, G) → s
  31. TriangleCenter(A, B, C, 3) → O
  32. Segment(C', A') → f_1
  33. Segment(A', B') → g_1
  34. Segment(B', C') → h_1
  35. TriangleCenter(A, B, C, 5) → N
  36. TriangleCenter(A, B, C, 175) → J
  37. TriangleCenter(A, B, C, 176) → K
  38. Circle(A, a) → c_1
  39. Circle(B, b) → d_1
  40. Circle(C, c) → e_1
  41. Intersect(c_1, e_1) → L, M, ,
  42. Line(M, L) → i_1
  43. Intersect(c_1, d_1) → Q, R, ,
  44. Line(Q, R) → j_1
  45. Intersect(d_1, e_1) → P, S, ,
  46. Line(S, P) → k_1
  47. Segment(A, Q) → l_1
  48. Segment(B, P) → m_1
  49. Segment(C, M) → n_1

AI Drawing Prompt Reference

Draw a triangle ABC. First, draw three ellipses related to vertices A, B, and C, and find their intersection points D and E, then connect D and E. Next, draw other intersection points F, G1, H1, I1 and connect corresponding line segments. Then, draw important triangle centers such as the Langechamps point, centroid, circumcenter, orthocenter, and find points A', B', C' on the Euler line. After that, draw perpendicular lines related to A', B', C'. Finally, draw the circumcircle and three circles related to vertices A, B, C, find their intersection points, and connect these intersection points. The whole figure illustrates the geometric relationships of triangle centers, the Euler line, and related circles.
The prompt above was compiled from the original drawing commands and is for reference only.