VMO 2013 Problem 3

Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$ . The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$ . $J$ is midpoint of $KL$ .a) Prove that $D,I,J$ collinear.b) $B,C$ are fixed points, $A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant. $IE,IF$ cut $(I)$ again at $M,N$ respectively. $MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.