The Pragmatick's answer is (currently) incorrect. Specifically, $AD$ (or $AC$) does not need to be the diameter.
Unlike in triangles, determining the side lengths of a quadrilateral does not guarantee a unique quadrilateral. The angle between $AB$ and $AC$ can vary, giving quadrilaterals that aren't cyclic.
If, however, you are wondering whether or not this quadirlateral is tangential, then The Pragmatick's first line is helpful. From the congruency we can tell the angle bisectors of the angle concur, so this would have to be the 'incenter'. More generally, in a quadrilateral if $AB + CD = AC + BD$, the quadrilateral is tangential.