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Incenter is Its Own Isogonal Conjugate

Incenter is Its Own Isogonal Conjugate
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2015-02-14 18:47
TriangleAngleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates that the incenter (I) is its own isogonal conjugate. Key mathematical concepts include the incenter of a triangle and isogonal conjugates. Using GeoGebra commands such as Line, AngularBisector, Intersect, OrthogonalLine, Circle, Segment, and Mirror, the construction of the incenter's isogonal conjugate is illustrated. Suitable for high school geometry learners and math enthusiasts.

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. Intersect(h, a) → E, K
  12. Intersect(h, c) → F, G
  13. Intersect(h, b) → H, J
  14. Segment(E, C) → i
  15. Segment(A, H) → j
  16. Segment(B, F) → k
  17. Intersect(i, j) → G_e
  18. Mirror(I, c) → I'_3
  19. Mirror(I, a) → I'_1
  20. Mirror(I, b) → I'_2
  21. Circle(I'_1, I'_3, I'_2) → p

AI Drawing Prompt Reference

Draw a triangle, mark its incenter and the three angle bisectors, then find the symmetric points of the incenter with respect to the triangle's sides, and finally connect these symmetric points to form a circle, demonstrating that the incenter is its own isogonal conjugate.
The prompt above was compiled from the original drawing commands and is for reference only.