OSN 2009 Problem 8

Given an acute triangle $ABC$ . The incircle of triangle $ABC$ touches $BC,CA,AB$ respectively at $D,E,F$ . The angle bisector of $\angle A$ cuts $DE$ and $DF$ respectively at $K$ and $L$ . Suppose $AA_1$ is one of the altitudes of triangle $ABC$ , and $M$ be the midpoint of $BC$ . (a) Prove that $BK$ and $CL$ are perpendicular with the angle bisector of $\angle BAC$ . (b) Show that $A_1KML$ is a cyclic quadrilateral.