|A quadrilateral is inscribed if and only if xy = ac + bd|
There is also the second Ptolemy relation :
|x/y = (ad + bc)/(ab + cd)|
Let's proof the direct theorem, that is when ABCD inscribed, then...
Recall the relations giving area of a triangle S = 1/2 b.c.sinA and S = a.b.c/(4R)
Area of quadrilateral is S = 1/2 x.y.sinθ (easy from area of the four internal triangles).
|xy = ac + bd|
Area(ABC) = a.b.x/(4R), Area(ACD) = c.d.x/(4R), Area(ABD) = a.d.y/(4R) and Area(BCD) = b.c.y/(4R) that is (ab + cd)x = (ad + bc)y and the second Ptolemy relation.