IMO 2000 - Shortlisted Problems Question 22

G3 (IND). Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D,E,F$ on sides $BC,CA,AB$ respectively such that $OD+DH=OE+EH=OF+FH$ and the lines $AD,BE,CF$ are concurrent.