ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90∘. E is the foot of the perpendicular from A to BC, and F is the foot of the perpendicular from A to DC. If AE+AF=16, what is the area of ABCD?

Answer 1

I think we are missing information here.
AB=AD, and ∠BAD=90° mean that BD is a diameter, and hence ∠BCD=90°.
Also, AB=AD means that ∠ABD=∠ADB=45°.
However, we do not know anything about ∠DBC or ∠BDC, just that they are complementary.
So ΔBDC is not defined.
If the question had been
What is the maximum area of ABCD, then we can say BC=CD=8,and hence solve the problem, area = 8*8=64.
As it is, I think information is still missing.