When you say CE=CF and BD=BF you are using the same principle you are trying to prove. That a point is equal distant from the two tangent points. How about:
angle AEO and angle ADO are right angles because a tangent is perpendicular to a radius
AO = AO by reflexive property
OD = OE =r
so triangle AOD is congruent to triangle AEO
by hypotenuse leg.
and AD = AE