Isogonal conjugation

Isogonal conjugation is a transformation related to a triangle. Given a triangle $ABC$ and a point $P$ (not on the sides), we can construct its isogonal conjugate $P'$ as follows: For each vertex (say $A$), draw the line $AP$. Then, reflect this line over the angle bisector at $A$. Do this for $B$ and $C$ as well. The three reflected lines will meet at a single point $P'$, which is called the isogonal conjugate of $P$ with respect to triangle $ABC$.

Important properties

• The isogonal conjugate of a point inside the triangle is also inside the triangle.

• The isogonal conjugate of the centroid is the symmedian point.

• Isogonal conjugation is an involution: the isogonal conjugate of the isogonal conjugate of $P$ is $P$ itself.

• Special triangle centers (like the incenter, circumcenter, orthocenter) have notable isogonal conjugates.