It's known that there is an isogonal conjugation with respect to triangle. For example, if $P$ is a point and $ABC$ is a triangle then isogonal conjugate of $P$ is defined as point such that $\angle PAB=\angle CAP'$, $\angle PBC=\angle ABP'$ and $\angle PCA=\angle BCP'$.

Analogously, we can define isogonal conjugation for quadrilateral (and even for $n$-gon). It's well-known that point $P$ has an isogonal conjugate with respect to quadrilateral $ABCD$ if and only if projections $P$ onto sides $ABCD$ lie on the circle (and similar criteria for $n$-gon).

However, I don't know proof of this fact and I want to learn more about this topic. Is there any article, book, etc., which can provide me more information about the topic?

Also it would be great if you could find examples of geometric problems which can be solved by this construction.


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