It is known that if four circles tangent cyclically to its neighbors,
than four tangency points are concyclic, i.e. lie on a circle.
But if we have four points on a given circle,
how to construct four circles, touching each other in these points?

There are primitive and general cases.
Steps of constructions are clear from demonstration.
In primitive case all four centers are fixed by positions of given (free-on-a-circle) points P,Q,R,S.
In general case one of this centers T are free - on - a - bissector of