ILL 1971 Problem 19

In a triangle $P_1P_2P_3$ let $P_iQ_i$ be the altitude from $P_i$ for $i = 1, 2,3$ ( $Q_i$ being the foot of the altitude). The circle with diameter $P_iQ_i$ meets the two corresponding sides at two points different from $P_i.$ Denote the length of the segment whose endpoints are these two points by $l_i.$ Prove that $l_1 = l_2 = l_3.$