Answer:
$\sqrt{3}$

Observe that $A E C F$ is a parallelogram. The equal area condition gives that $B E=D F=\frac{1}{3} A B$. Let $C E \cap B D=X$, then $\frac{E X}{C X}=\frac{B E}{C D}=\frac{1}{3}$, so that $B X^{2}=E X \cdot C X=3 E X^{2} \Rightarrow B X=\sqrt{3} E X \Rightarrow$ $\angle E B X=30^{\circ}$. Now, $C E=2 B E=C F$, so $C E F$ is an equilateral triangle and $C D=\frac{3}{2} C F=\frac{3}{2}$. Hence, $B D=\frac{2}{\sqrt{3}} \cdot \frac{3}{2}=\sqrt{3}$.
2. In rectangle $A B C D$, points $E$ and $F$ lie on sides $A B$ and $C D$ respectively such that both $A F$ and $C E$ are perpendicular to diagonal $B D$. Given that $B F$ and $D E$ separate $A B C D$ into three polygons with equal area, and that $E F=1$, find the length of $B D$.