KöMaL 2024 C1807

In triangle $ABC$ angles satisfy $2\beta = 3\gamma$. Let points $D$ and $E$ be chosen on side $AC$ such that $BD$ and $BE$ trisect angle $\beta$, and point $D$ is located between points $A$ and $E$. Further, let $F$ denote the intersection of side $AB$ and the angle bisector of $\gamma$. Prove that $BE$ and $DF$ are parallel.