AMC 12 2006a Problem 17

Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. What is $r/s$?

unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=3;
pair B=(0,0), C=(3,0), D=(3,3), A=(0,3);
pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6);
pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0];
pair[] dots={A,B,C,D,Ep,F};
draw(A--F);
draw(Circle(Ep,5/3));
draw(A--B--C--D--cycle);
dot(dots);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",Ep,E);
label("$F$",F,NW);

$\mathrm{(A) \ } \frac{1}{2} \mathrm{(B) \ } \frac{5}{9} \mathrm{(C) \ } \frac{3}{5} \mathrm{(D) \ } \frac{5}{3} \mathrm{(E) \ } \frac{9}{5}$