NL-NWO 1980 Problem 3

Given is the non-right triangle $ABC$ . $D,E$ and $F$ are the feet of the respective altitudes from $A,B$ and $C$ . $P,Q$ and $R$ are the respective midpoints of the line segments $EF$ , $FD$ and $DE$ . $p \perp BC$ passes through $P$ , $q \perp CA$ passes through $Q$ and $r \perp AB$ passes through $R$ . Prove that the lines $p, q$ and $r$ pass through one point.