Challenge · Medium problem

In $\triangle ABC$, points $D$, $E$, $F$ lie on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively with $BD:DC=CE:EA=AF:FB=1:2$. Cevians $\overline{AD}$, $\overline{BE}$, $\overline{CF}$ bound a small central triangle. Find the ratio of the area of this central triangle to the area of $\triangle ABC$.

Express your answer as a common fraction.