Because AB is the diameter, CE⊥AB

So AB bisects CE vertically,

So ∠COM=∠EOM,

In △CDE, ∠FDE=∠DCE+∠DEC,

And ∠DCE=∠DOE/2, ∠DEC=∠DOC/2,

So ∠DCE+∠DEC=(∠DOE+∠DOC)/2=∠EOM.

So ∠FDE=∠EOM

And ∠FMD=∠EMO (equal to the vertex angle)

So △FMD∽△EMO

So FM/ME=DF/OE

So FM×OE=DF×ME.

So FM×OB=DF×MC.