MathPlotAI Logo MathPlotAI

Gergonne Point and Nagel Point as Isotomic Conjugates

Gergonne Point and Nagel Point as Isotomic Conjugates
Loading the math board and drawing, please wait…
2015-02-14 21:27
TriangleAngleGeometric Construction
In collections A Tour of Triangle Geometry
This example illustrates the isotomic conjugate relationship between the Gergonne point and the Nagel point. By constructing the Gergonne point G and the Nagel point N, and their projections on side BC, the geometric properties of these special points are demonstrated. GeoGebra commands used include line, angular bisector, intersection, orthogonal line, circle, polygon, segment, midpoint, etc. Suitable for high school geometry learners and educators.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. AngularBisector(B, A, C) → d
  5. AngularBisector(C, B, A) → e
  6. AngularBisector(A, C, B) → f
  7. Intersect(d, e) → I
  8. OrthogonalLine(I, c) → g
  9. Intersect(c, g) → D
  10. Circle(I, D) → h
  11. Circle(B, A, C) → p
  12. OrthogonalLine(A, d) → i
  13. OrthogonalLine(B, e) → j
  14. Intersect(f, i) → E
  15. Intersect(e, i) → F
  16. Intersect(d, j) → G_1
  17. OrthogonalLine(E, c) → k
  18. Intersect(c, k) → H
  19. Circle(E, H) → q
  20. OrthogonalLine(F, b) → m
  21. Intersect(b, m) → J
  22. Circle(F, J) → r
  23. OrthogonalLine(G_1, a) → n
  24. Intersect(a, n) → K
  25. Circle(G_1, K) → s
  26. Intersect(r, c, 1) → B'
  27. Intersect(q, a, 1) → C'
  28. Intersect(s, b, 1) → N
  29. Polygon(N, C', B') → poly1, l_1, n_1, m_1
  30. Segment(B, B') → t
  31. Segment(A, N) → a_1
  32. Segment(C', C) → b_1
  33. Intersect(t, a_1) → N_a
  34. Intersect(h, c, 2) → L
  35. Intersect(h, a, 1) → M_1
  36. Intersect(h, b, 1) → G
  37. Segment(A, G) → c_1
  38. Segment(M_1, C) → d_1
  39. Segment(L, B) → e_1
  40. Intersect(c_1, d_1) → G_e
  41. Polygon(L, M_1, G) → poly2, n_2, l_2, m_2
  42. Midpoint(B, C) → M