Problem

Source: Iran MO 2018, second round, day 2, P6

Tags: geometry, Iran 2nd Round



Two circles $\omega_1,\omega_2$ intersect at $P,Q $. An arbitrary line passing through $P $ intersects $\omega_1 , \omega_2$ at $A,B $ respectively. Another line parallel to $AB $ intersects $\omega_1$ at $D,F $ and $\omega_2$ at $E,C $ such that $E,F $ lie between $C,D $.Let $X\equiv AD\cap BE $ and $Y\equiv BC\cap AF $. Let $R $ be the reflection of $P $ about $CD$. Prove that: a. $R $ lies on $XY $. b. PR is the bisector of $\hat {XPY}$.