Isosceles Twins, part 2

$\triangle ABC$ is an isosceles triangle with a base length of 1. $AD \perp CB$ and $ED \perp AC$ . $E$ is chosen so that the two incircles are congruent. If their radius is a root of $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ , where $a>0$ and $\gcd (a,b,c,d,e) = 1$ , find $f(2)$ .