OMCC 2005 Problem 3

Let $ABC$ be a triangle. $P$ , $Q$ and $R$ are the points of contact of the incircle with sides $AB$ , $BC$ and $CA$ , respectively. Let $L$ , $M$ and $N$ be the feet of the altitudes of the triangle $PQR$ from $R$ , $P$ and $Q$ , respectively.

a) Show that the lines $AN$ , $BL$ and $CM$ meet at a point.

b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle $PQR$ .

Aarón Ramírez, El Salvador