OMM 1999 Problem 3

A point $P$ is given inside a triangle $ABC$ . Let $D,E,F$ be the midpoints of $AP,BP,CP$ , and let $L,M,N$ be the intersection points of $BF$ and $CE, AF$ and $CD, AE$ and $BD$ , respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$ . (b) Prove that $DL,EM$ , and $FN$ are concurrent.