OBM 2009 Problem 5

Let $ABC$ be a triangle and $O$ its circumcenter. Lines $AB$ and $AC$ meet the circumcircle of $OBC$ again in $B_1\neq B$ and $C_1 \neq C$ , respectively, lines $BA$ and $BC$ meet the circumcircle of $OAC$ again in $A_2\neq A$ and $C_2\neq C$ , respectively, and lines $CA$ and $CB$ meet the circumcircle of $OAB$ in $A_3\neq A$ and $B_3\neq B$ , respectively. Prove that lines $A_2A_3$ , $B_1B_3$ and $C_1C_2$ have a common point.