RO-NMO 2011 Problem 93

Let $ABC$ be a triangle, $I_a$ be center of the $A\text{-excircle}.$ This excircle intersects the lines $AB, AC,$ at $P,$ respectively, $Q.$ The line $PQ$ intersects the lines $I_aB,I_aC$ at $D,$ respectively, $E.$ Let $A_1$ be the intersection of $DC$ with $BE,$ and define, analogously, $B_1,C_1.$ Show that $AA_1,BB_1,CC_1$ are concurrent.