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 BE. Point F lies on the circle, on the same side of BE as A. Segment AF is tangent to the circle, and AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}. What is r/s?
[asy]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);[/asy]<spanclass=′latex−bold′>(A)</span>21<spanclass=′latex−bold′>(B)</span>95<spanclass=′latex−bold′>(C)</span>53<spanclass=′latex−bold′>(D)</span>35<spanclass=′latex−bold′>(E)</span>59 trigonometryquadraticsPythagorean Theoremgeometrytrig identitiesLaw of Cosinesalgebra