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Isogonal Conjugate of a Point on the Circumcircle

Isogonal Conjugate of a Point on the Circumcircle
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2015-02-14 21:39
CircleAngleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the isogonal conjugate of a point on the circumcircle, which is at infinity. Key concepts include isogonal conjugation and the point at infinity. GeoGebra commands used include Line, Circle, Mirror, and Center. Suitable for high school geometry.

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. Mirror(P, c) → P'_3
  6. Mirror(P, a) → P'_1
  7. Mirror(P, b) → P'_2
  8. Circle(P'_1, P'_3, P'_2) → e
  9. Center(e) → P'

AI Drawing Prompt Reference

Draw a triangle and its circumcircle. Take a point on the circumcircle and draw perpendicular lines from this point to each side of the triangle, meeting the sides at three points. Circle these three points. Then, reflect the original point across each perpendicular line to get three new points. Finally, draw a circle using these three new points as centers and find the center of this circle.
The prompt above was compiled from the original drawing commands and is for reference only.