ISL 1991 Problem 3

Let $S$ be any point on the circumscribed circle of $PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $l(S, PQR).$ Suppose that the hexagon $ABCDEF$ is inscribed in a circle. Show that the four lines $l(A,BDF),$ $l(B,ACE),$ $l(D,ABF),$ and $l(E,ABC)$ intersect at one point if and only if $CDEF$ is a rectangle.