Question 80
Let $\triangle ABC$ 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.