The condition is:
$c\sin x+a\sin y=b\sin (x+y)$.
Proof:
Name the points $O,A,B,C$ so that $OA=a,OB=b,OC=c$. Perform an inversion with center $O$ and radius $b$; clearly this transformation fixes $B$. Say, this sends the points $A,B,C$ to $P,Q,R$ respectively (we already know that $Q\equiv B$), and define $p,q,r$ by $OP=p$ and so on. The properties of inversion imply that $O,A,B,C \text{ are concyclic} \iff P,Q,R \text{ are collinear}.$ Then taking $OQ=OB$ as the $x-$axis, the coordinates of these points are $P(p\cos x,p\sin x),Q(q,0),R(r\cos y,-r\sin y).$ So the condition $P,Q,R$ being collinear translates to $$\det\left| \begin{array}{ccc} q & 0 & 1\\ p\cos x & p\sin x &1\\ r\cos y & -r\sin y & 1\end{array}\right|= 0$$ and after some simple calculations, this reduces to $q(p\sin x+r\sin y)=pr \sin(x+y)$. Now note that by the definition of inversion, $p=\frac{b^2}{a},q=b,r=\frac{b^2}{c}$; so subbing these into the previous equation yields the desired result : $$c\sin x+a\sin y=b\sin (x+y).$$ $\blacksquare$