John Conway <conway@math.princeton.edu> writes:

... Alternatively, let AEPF be any circle through A and P, and then BFPD and CEPD be the circumcircles of BFP and CEP (which will automatically intersect at a point D on BC).

This is all part of "the Miquel theory" of the triangle. JHC

By considering the point at infinity, I understand this to mean that if we have points 01234567 in cocircular quartets 0123, 4567, 0145, 2367, and 0246, then quartet 1357 must also be cocircular. I didn't know that. Are all such patterns known? Say, given a mapping of a hypergraph to the plane, where vertices are mapped to points, and the images of the vertices of each hyperedge are cocircular, can we tell what other subsets of vertices are forced to map to cocircular points? Dan