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Isogonal Conjugate of a Line

Isogonal Conjugate of a Line
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2015-02-15 08:40
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the isogonal conjugate of a line, a special circumconic passing through the vertices of a triangle. If the line intersects the circumcircle at two real points, its isogonal conjugate is a circumconic through the triangle's vertices. Constructed using GeoGebra commands such as Line, Circle, Intersect, Polygon, etc. Suitable for high school geometry education.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Center(d) → O
  6. OrthogonalLine(A, b) → l
  7. OrthogonalLine(B, c) → m
  8. Intersect(l, m) → H
  9. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  10. Line(X, Y) → n
  11. Point(n) → P
  12. Mirror(P, a) → M'
  13. Mirror(P, c) → M'_1
  14. Mirror(P, b) → M'_2
  15. Circle(M', M'_1, M'_2) → e
  16. Center(e) → D
  17. Locus(D, P) → loc1

AI Drawing Prompt Reference

Draw a triangle and its circumcircle. Then, construct perpendicular lines from each vertex to the adjacent sides, finding their intersections. Use these intersections as vertices to form a new triangle. Finally, draw a circle passing through the vertices of this new triangle, which represents the isogonal conjugate of the original line.
The prompt above was compiled from the original drawing commands and is for reference only.