, 2019

Let $P$ be a point inside the acute triangle $ABC$, and let $Q$ be the isogonal conjugate of $P$. Let $L$, $M$ and $N$ be the midpoints of the shorter arcs $BC$, $CA$ and $AB$ of the circumcircle of $ABC$, respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $PBC$, let $X_B$ be the intersection of ray $MQ$ and circle $PCA$, and let $X_C$ be the intersection of ray $NQ$ and circle $PAB$. Prove that $P$, $X_A$, $X_B$ and $X_C$ are concyclic or coincide.