Let's say the length of rectangle ABCD is L, and the width is W. If r is the radius of circle E, then the length of rectangle EFDG is (L-r) and the width is (W-r).
We can split triangle ABC into three smaller triangles: AEC, AEB and BEC. Each has an altitude of r.
area(ABC) = area(AEC) + area(AEB) + area(BEC)
So the area of rectangle EFDG is
Whoa. So I suppose unless we are provided with the L/W ratio, we will not have a pretty answer for this problem.