Thank you! Clifford's chain of theorems--the very last item in ch 14. (I now surely remember you writing up something so named.) Coxeter says: Put four arbitrary black circles through a point. Set aside each in turn. For each of the four remaining triads, put a red circle through their three second intersections. The four red circles meet at a point, and red and black can be interchanged. Now put a 5th black circle through the first point. Set aside each black circle in turn. Each remaining tetrad determines a quadruple red concurrence, as before. The five concurrences lie on a circle ... --rwg On 2016-01-26 12:49, Eugene Salamin via math-fun wrote:

These theorems come out of Coxeter, "Introduction to Geometry", 2nd ed., sec. 14.8-14.9.

  --  Gene

From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Sent: Tuesday, January 26, 2016 7:55 AM Subject: [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