Let the points be $A,B,C,D$ such that $C$ and $D$ lie on the same side of line $AB$. Suppose WLOG that $\angle ACB\ge\angle ADB$, hence point $C$ is either on the circumference or inside the circle $(ADB)$.

We consider the full convex of $A,B,C,D$. Case1:full convex is triangle Circumscribed circle of triangle satisfies the condition. Case2:full convex is quadrilateral Let full convex be $ABCD$.By PHP,$A+C\ge 180^{\circ}$ or $B+D\ge 180^{\circ}$.WLOG $A+C\ge 180^{\circ}$,then circumscribed circle of $BCD$ satisfies the condition.