IR-TST 2018 Problem 18

Consider quadrilateral $ABCD$ inscribed in circle $\omega$ . $P\equiv AC\cap BD$ . $E$ , $F$ lie on sides $AB$ , $CD$ respectively such that $\hat {APE}=\hat {DPF}$ . Circles $\omega_1$ , $\omega_2$ are tangent to $\omega$ at $X$ , $Y$ respectively and also both tangent to the circumcircle of $\triangle PEF$ at $P$ . Prove that: $$\frac {EX}{EY}=\frac {FX}{FY}$$ Proposed by Ali Zamani