Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.