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Mathematics Examination Paper of Hangzhou Senior High School Entrance Examination in 2008
O 1 not funny, I use F.

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