Local Mathematical Competitions

All poles and polars are considered with respect to the given circumcircle of $ABCD$. To start with, notice that line $q=PR$ is the polar of $Q$ and line $p=QR$ is the polar of $P$. As $Y$ lies on the polar of $P$, it follows that $P$ lies on the polar $y$ of $Y$. The pole of the line $OY$ is the point at infinity $\infty$ on the direction $BC$, as $OY$ is a diameter line. Thus the polar $y$ of $Y$ is the line $P\infty =BC$, implying that $YB$ is tangent to the given circle at $B$. Similar considerations show that $XB$ is tangent to the given circle at $B$, hence proving the thesis.