Question 124
$A$ is the center of a semicircle, with radius $A D$ lying on the base. $B$ lies on the base between $A$ and $D$, and $E$ is on the circular portion of the semicircle such that $E B A$ is a right angle. Extend $E A$ through $A$ to $C$, and put $F$ on line $C D$ such that $E B F$ is a line. Now $E A = 1$, $A C = \sqrt{2}$, $B F = \frac{2 - \sqrt{2}}{4}$, $C F = \frac{2 \sqrt{5} + \sqrt{10}}{4}$, and $D F = \frac{2 \sqrt{5} - \sqrt{10}}{4}$. Find $D E$.