Question 50
Consider quadrilateral $ABCD$ inscribed in circle $\omega$. $P$ is the intersection point of $AC$, $BD$. Points $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\widehat{APE} = \widehat{DPF}$. 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
$$
\frac{EX}{EY} = \frac{FX}{FY}.
$$
$$
\frac{EX}{EY} = \frac{FX}{FY}.
$$