Let four circles (blue) pass through a single point, M. Each pair of these circles intersect at a second point (pink). Each three of the four blue circles will have three pink points among them; these trios of pink points define four new circles (brown), which intersect in a single point, P.
If we start with five circles passing through a single point M, then we can apply the procedure above to each subset of four of them. This will produce five points P that all lie on a single circle.
If we start with six circles that all pass through a single point M, then each subset of five of them defines a new circle, as we’ve just seen. These six new circles all pass through a single point.
Remarkably, this pattern continues forever. It was discovered by the English geometer William Kingdon Clifford.