MathNet 2011 Question 43

Let $ABC$ be a triangle with $AB\gt BC$. Let $D$ be the midpoint of $AC$, and let $E$ be the intersection of the bisector of $\angle ABC$ with the line $AC$. Let $F$ lie on $BE$ with $CF\perp BE$, and let $G=CF\cap BD$.
Prove that $DF$ bisects the segment $EG$.