ILL 1987 Problem 47

Through a point $P$ within a triangle $ABC$ the lines $l, m$ , and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$ , $m$ intersects $AC$ in $R$ , and $n$ intersects $AB$ in $S$ , then the points $Q, R$ , and $S$ are collinear.