Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and let $G$ be the point of the segment $AD$ such that $AG = 2GD$. Let $E$ and $F$ be points on the sides $AB$ and $AC$, respectively, such that$ G$ lies on the segment $EF$. Let $M$ and $N$ be points of the segments $AE$ and $AF$, respectively, such that $ME = EB$ and $NF = FC$. a) Prove that the area of the quadrilateral $BMNC$ is equal to four times the area of the triangle $DEF$. b) Prove that the quadrilaterals $MNFE$ and $AMDN$ have the same area.