AE^2=AO^2+OF^2=(2r)^2+r^2

AE=√[(2r)^2+r^2]

Connect FD, so FD is perpendicular to AE.

Angle Dao = 2 times angle FAD = 60, so angle AEO = 30.

So in Rt triangle DEF,

DE=√3*FD=√3*r