Are you familiar with the angle bisector theorm? Well, by this theorm, once we recognize AC as the angle bisector, we can set up a proportion: .
Since we have DC=BC (let's call these two sides x) we have
Now, by definition, we have ABCD as a kite. Look up some properties of a kite, and see if you can go on to prove that it is a cyclic quadrilateral. (Hint: It will be cyclic if it is formed by two right triangles)