Question 134
The points $A_1$, $B_1$, $C_1$ are chosen on the sides $BC$, $CA$, $AB$ of a triangle $ABC$ so that $BA_1 = BC_1$ and $CA_1 = CB_1$. The lines $C_1A_1$ and $A_1B_1$ meet the line through $A$, parallel to $BC$, at $P$, $Q$. Let the circumcircles of the triangles $APC_1$ and $AQB_1$ meet at $R$. Given that $R$ lies on $AA_1$, show that $R$ lies on the incircle of $ABC$.
(Emil Kolev)
(Emil Kolev)