Autumn tournament

The points $A_1$, $B_1$, $C_1$ are chosen on the sides $BC$, $CA$, $AB$ of a triangle $ABC$ so that $B{A}_1=B{C}_1$ and $C{A}_1=C{B}_1$. The lines $C_1{A}_1$ and $A_1{B}_1$ meet the line through $A$, parallel to $BC$, at $P$, $Q$. Let the circumcircles of the triangles $AP{C}_1$ and $AQ{B}_1$ meet at $R$. Given that $R$ lies on $A{A}_1$, show that $R$ lies on the incircle of $ABC$. (Emil Kolev)