• Let $S_a = \frac{AE}{AB} + \frac{AF}{AC}$. Define $S_b,S_c$ similarly.
• Use pigeon hole principle.
• Let $M_a = \frac{AE}{AB} \times \frac{AF}{AC}$. Define $M_b,M_c$ similarly.
• Use AM-GM inequality to find relation between $S_x$ and $M_x$ for some $x$. If you don't know about AM-GM inequality, you can google about it.
• The area of $\triangle ABC=\frac{\sin{\angle EBD}}{2}\times AB \times BC$

Let $S_a=\frac{AE}{AB} + \frac{FA}{CA}, S_b= \frac{EB}{AB} + \frac{BD}{BC}, S_c=\frac{DC}{BC} + \frac{CF}{CA}$.

Let $M_a=AE/AB \times FA/CA, M_b=EB/AB \times BD/BC, M_c=DC/BC \times CF/CA$.

Now, 

$S_a + S_b + S_c $

$= \frac{AE}{AB} + \frac{FA}{CA} + \frac{EB}{AB} + \frac{BD}{BC} + \frac{DC}{BC} + \frac{CF}{CA} $

$= 3$.

Without loss of generality, we assume that $S_b$ is the lowest among $S_a,S_b,S_c$. By pigeon hole principle, we know that $S_b \leq 1$.

From AM-GM inequality, we know that $\frac{S_b}{2} \geq \sqrt{M_b}$ or $\frac{1}{4} \geq \frac{S_b}{4} \geq M_b$.

Now, the area of $\triangle BED$

$ = \frac{\sin{\angle EBD}}{2}\times BE \times BD $

$= \frac{\sin{\angle ABC}}{2}\times AB \times BC\times M_b $

$= 2016\times M_b \leq 2016 \times \frac{1}{4} $

$= 504$.