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Isogonal Conjugates Proof (Symmedians)

Isogonal Conjugates Proof (Symmedians)
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2020-06-02 11:57
TriangleAngleVectorGeometric Construction
This example demonstrates the existence of isogonal conjugates through the construction of symmedians. It involves core concepts like isogonal conjugates, symmedians, angle bisectors, reflections, line intersections, and perpendicular lines. The construction uses GeoGebra commands such as Polygon, AngularBisector, Mirror, Line, Intersect, OrthogonalLine, and Segment. Suitable for high school and advanced learners exploring geometric constructions and proofs.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. AngularBisector(B, A, C) → i
  3. AngularBisector(A, C, B) → j
  4. AngularBisector(C, B, A) → k
  5. Mirror(D, j) → D_1
  6. Mirror(D, i) → D_2
  7. Mirror(D, k) → D_3
  8. Line(C, D_1) → l
  9. Line(A, D_2) → m
  10. Line(B, D_3) → n
  11. Intersect(l, m) → S
  12. OrthogonalLine(B, b) → p
  13. Intersect(p, b) → F
  14. OrthogonalLine(A, a) → q
  15. OrthogonalLine(C, c) → r
  16. Intersect(r, c) → G
  17. Intersect(q, a) → E
  18. Polygon(F, G, E) → t2, g_1, e, f_1
  19. Intersect(m, a) → X
  20. Intersect(b, n) → Y
  21. Intersect(c, l) → Z
  22. Segment(D, E) → s
  23. Segment(D, G) → t
  24. Segment(D, F) → d
  25. Line(C, D) → h
  26. Intersect(h, f_1) → N
  27. Line(A, D) → g
  28. Intersect(g, g_1) → L
  29. Line(B, D) → f
  30. Intersect(f, e) → M
  31. Intersect(q, p) → H
  32. Intersect(i, k) → I
  33. Segment(D, D_2) → h_1
  34. Segment(D, D_3) → i_1
  35. Segment(D, D_1) → j_1