VN-TST 2003 Problem 2

Given a triangle $ABC$ . Let $O$ be the circumcenter of this triangle $ABC$ . Let $H$ , $K$ , $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$ , $B$ , $C$ , respectively. Denote by $A_{0}$ , $B_{0}$ , $C_{0}$ the midpoints of these altitudes $AH$ , $BK$ , $CL$ , respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ , $CA$ , $AB$ at the points $D$ , $E$ , $F$ , respectively. Prove that the four lines $A_{0}D$ , $B_{0}E$ , $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$ , we consider the line $OI$ as an arbitrary line passing through $O$ .)