Question 108
Let $ABC$ be a scalene triangle with $BC\gt CA\gt AB$. Circles $\omega,\gamma$ pass through $A$ and have centers $B,C$, respectively. They meet segment $BC$ at $M,N$, respectively. Let $Z$ be the reflection of $A$ in the midpoint of $MN$.
(a) Let $P=ZM\cap AC$. Prove triangle $CPM$ is isosceles.
(b) Let $X\ne M$ be the second intersection of $ZM$ with $\omega$. Prove $BX\parallel AC$.
(c) Let $Y\ne N$ be the second intersection of $ZN$ with $\gamma$. Prove $A,X,Y$ are collinear.
(a) Let $P=ZM\cap AC$. Prove triangle $CPM$ is isosceles.
(b) Let $X\ne M$ be the second intersection of $ZM$ with $\omega$. Prove $BX\parallel AC$.
(c) Let $Y\ne N$ be the second intersection of $ZN$ with $\gamma$. Prove $A,X,Y$ are collinear.