MathNet 2022-part-05 — 197 problems (EN–VI) Question 63

Let $\omega$ be the circumcircle of triangle $ABC$, and let $\Omega_A$ be its excircle tangent to side $BC$. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ be the projection of $A$ onto the tangent to $\Omega_A$ at $X$, and let $Q$ be the projection of $A$ onto the tangent to $\Omega_A$ at $Y$. The tangent at $P$ to the circumcircle of triangle $APX$ and the tangent at $Q$ to the circumcircle of triangle $AQY$ meet at a point $R$.
Prove that the lines $AR$ and $BC$ are perpendicular.