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Send one of the points to infinity by a Mobius tranformation. Your set of points now has the property that the intersection of any two lines passing from pairs of points in the set is also in the set. Such configurations are either all collinear, dense in the plane, or one of these two exceptional cases:

• A point together with a collection of points in a line
• The four vertices of a parallelogram and the intersection of its diagonals.

To prove your result we need to rule out the first case. This can be done by considering circumcircles of triangles with disjoint bases and the common apex. For the second case one can check that it is circularly stable only when the parallelogram is inscribed in a circle, i.e. it is a rectangle. So the maximum size of a finite circularly stable set is 6 (The four verties and center of a rectangle and the point at infinity).

The result I used above is proved in "A dense planar point set from iterated line intersections" by D. Ismailescu and R. Radoicic.