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).