TR-MO 1994 Problem 6

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$ . Let $K$ be the point on $CB$ with $CK=BD$ , and $L$ be the point on $CA$ with $AE=CL$ . Lines $AK$ and $BL$ meet at $P$ . If $Q$ is the midpoint of $BC$ , $I$ the incenter, and $G$ the centroid of $\triangle ABC$ , show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.