[math-fun] circles though points

I think it was Gene Salamin, in an proposed HAKMEM sequel, who described a remarkable infinite sequence of geometry theorems that went something like: Put three arbitrary circles through a given point. They intersect at three other points which (obviously) lie on a 4th circle. Add an arbitrary(?) (5th) point. Put circles through the four triplets of points formed by lumping this point with two of the three intersections. These four circles intersect at one point. Invent a 6th point, and repeat this whole process, to create five quadruple intersections. These all lie on one circle. Iterate, alternately getting n circles through a point, or n points on a circle. Can someone correct and identify this half-remembered theorem? --rwg