USAMO 1995 Problem 3

The circumcenter $\mathrm{O}$ of the triangle $\mathrm{ABC}$ does not lie on any side or median. Let the midpoints of $\mathrm{BC}, \mathrm{CA}$, $\mathrm{AB}$ be $\mathrm{L}, \mathrm{M}, \mathrm{N}$ respectively. Take $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ on the rays $\mathrm{OL}, \mathrm{OM}, \mathrm{ON}$ respectively so that $\angle \mathrm{OPA}=\angle \mathrm{OAL}$, $\angle \mathrm{OQB}=\angle \mathrm{OBM}$ and $\angle \mathrm{ORC}=\angle \mathrm{OCN}$. Show that $\mathrm{AP}, \mathrm{BQ}$ and $\mathrm{CR}$ meet at a point.