2021 IMO Shortlist G8

Let $\omega$ be the circumcircle of a triangle $A B C$, and let $\Omega_{A}$ be its excircle which is tangent to the segment $B C$. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_{A}$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_{A}$ at $X$ and $Y$, respectively. The tangent line at $P$ to the circumcircle of the triangle $A P X$ intersects the tangent line at $Q$ to the circumcircle of the triangle $A Q Y$ at a point $R$. Prove that $A R \perp B C$.

令 $\omega$ 为三角形 $A B C$ 的外接圆，并令 $\Omega_{A}$ 为其与线段 $B C$ 相切的外圆。设$X$和$Y$为$\omega$和$\Omega_{A}$的交点。设$P$和$Q$分别为$A$在$X$和$Y$处到$\Omega_{A}$的切线的投影。三角形$A P X$ 外接圆的$P$ 切线与三角形$A Q Y$ 外接圆$Q$ 的切线相交于点$R$。证明$A R \perp B C$。