Question 108
Two circles $\omega_1$ and $\omega_2$ intersect each other at points $P$ and $Q$. An arbitrary line passes through $P$ and intersects $\omega_1$ and $\omega_2$ respectively at points $A$ and $B$. A line parallel to $AB$ cuts $\omega_1$ at $D$ and $F$, and cuts $\omega_2$ at $C$ and $E$ in a way that $E$ and $F$ lie between $C$ and $D$. Let $X$ be the intersection point of $BE$ and $AD$, $Y$ be the intersection of $AF$ and $BC$, and $R$ be the reflection of point $P$ over $CD$.
i)
Prove that $R$ lies on $XY$.
ii)
Prove that $PR$ is the angle bisector of $\angle XPY$.
i)
Prove that $R$ lies on $XY$.
ii)
Prove that $PR$ is the angle bisector of $\angle XPY$.