US-TST 2017 Problem 5

Let $A B C$ be a triangle with altitude $\overline{A E}$. The $A$-excircle touches $\overline{B C}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $D G$ and $D F$ such that quadrilateral $E V A N$ is a rhombus.