A Dot Product of Two Feet

In an acute triangle $ABC$, $\tan A=\dfrac 12$, and $D$ is a point on $BC$ such that $\triangle ABD$ and $\triangle ACD$ have areas $2$ and $4$ respectively. Drop perpendiculars $DE\perp AB$ at $E$ and $DF\perp AC$ at $F$. Find $\overrightarrow{DE}\cdot\overrightarrow{DF}$.