VTMO 2017 — Grade 7 — final Question 5

$ABC$ is isosceles at $A , BD = \frac{BC}{3}$. $E$ is beyond $D$ on $AD$ with $DE = AD$; $F$ is beyond $E$ on $BE$ with $EF = BE$. Prove $FD$ perpendicular to $BC$, and find $FD$ for equilateral $ABC$ with side $1$.