Challenge · Medium problem

In $\triangle ABC$, points $D$, $E$, $F$ lie on sides $BC$, $CA$, $AB$ respectively with $\dfrac{BD}{DC}=\dfrac{CE}{EA}=\dfrac{AF}{FB}=\dfrac{1}{2}$. The cevians $AD$, $BE$, $CF$ bound a central triangle. Express the area of this central triangle as a fraction of the area of $\triangle ABC$.

Express your answer as a common fraction.