Pentagonal Circles

$ABCDE$ is a regular pentagon with side length = 1 drawn on the cartesian plane with $A = (-\frac{1}{2},0)$ and $B=(\frac{1}{2},0)$ . Side $DC$ has been extended to $G$ and $F$ has been chosen so that the incircles of $\triangle ABF$ and $\triangle FGC$ are congruent. If the x-coordinate of $F$ is a root of the polynomial with integer coefficients $f(x) = ax^4+bx^3+cx^2+dx+e$ , where $a>0$ and $\gcd (a,b,c,d,e) = 1$ , find $f(0)$ .