2019 IMO Shortlist G3

In triangle $A B C$, let $A_{1}$ and $B_{1}$ be two points on sides $B C$ and $A C$, and let $P$ and $Q$ be two points on segments $A A_{1}$ and $B B_{1}$, respectively, so that line $P Q$ is parallel to $A B$. On ray $P B_{1}$, beyond $B_{1}$, let $P_{1}$ be a point so that $\angle P P_{1} C=\angle B A C$. Similarly, on ray $Q A_{1}$, beyond $A_{1}$, let $Q_{1}$ be a point so that $\angle C Q_{1} Q=\angle C B A$. Show that points $P, Q, P_{1}$, and $Q_{1}$ are concyclic. (Ukraine)

在三角形$A B C$中，设$A_{1}$和$B_{1}$为边$B C$和$A C$上的两点，设$P$和$Q$分别为线段$A A_{1}$和$B B_{1}$上的两点，则线$P Q$与$A B$平行。在射线$P B_{1}$上，超出$B_{1}$，令$P_{1}$为一点，使得$\angle P P_{1} C=\angle B A C$。类似地，在射线$Q A_{1}$上，超出$A_{1}$，令$Q_{1}$为一个点，使得$\angle C Q_{1} Q=\angle C B A$。证明点 $P、Q、P_{1}$ 和 $Q_{1}$ 是同循环的。 (乌克兰)