Problem #WSP-000233

Problem

Let $ABCDEF$ be a regular hexagon. Points $G$ and $H$ lie on $EF$ and $DE$ respectively such that $|EG|=|EH|$. Furthermore, the area of quadrilateral $ABGF$ is equal to the area of quadrilateral $BGEH$, which are both equal to the area of $BCDH$. What’s the ratio $\frac{|EG|}{|EF|}$?