SHARYGIN 2010 Problem 19

A quadrilateral $ABCD$ is inscribed into a circle with center $O$. Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drown to that circle at these points meet the line $AB$ in points $E$ and $F$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$. Prove that $XV = YU$.