Recently I stumbled upon Evan’s Isogonal Conjugates post on WordPress, and I have a couple of theorems that I feel are worth adding along to his post to my own blog.
1. Let Q be the isogonal conjugate of P with respect to ABC. Let AP,AQ intersect BC at X,Y, respectively. Then (BX*BY)/(CX*CY)=c^2/b^2.
Proof: Use areas (I haven’t actually done this) or Sine Law (I know this proof is slick).
2. Consider ABC and point P with isogonal conjugate Q, and let P have pedal triangle DEF. Then AQ⟂EF.
Proof: Use right triangles and the trivial isogonal angle equality theorem.
You can use this to provide a “brain-dead” proof for Theorem 3 (Pedal Circles).
A final note: Pascal’s Theorem is true for any conic and an inscribed hexagon, though the proof is pretty much identical. This is because you can project a conic to an ellipse, and project/dilate it to a circle, where the proof proceeds identically!
Evan’s post is here: https://usamo.wordpress.com/2014/11/30/three-properties-of-isogonal-conjugates/