TUYMAADA 2009 S7

A triangle $ABC$ is given. Let $B_1$ be the reflection of $B$ across the line $AC$ , $C_1$ the reflection of $C$ across the line $AB$ , and $O_1$ the reflection of the circumcentre of $ABC$ across the line $BC$ . Prove that the circumcentre of $AB_1C_1$ lies on the line $AO_1$ .

Proposed by A. Akopyan