On 9/26/06, Fred lunnon <fred.lunnon@gmail.com> wrote:
I'm afraid I have to withdraw the "Seven Circle Theorem" as a contender for this list. When I sat down to examine the proof, I realised it is actually the "Eight Circle Theorem", and ought to be rendered
Let A,C,D be circles tangent to one circle P, and B,E,F circles tangent to another circle Q [which may coincide with P or not]. If A,B,C,D,E,F,A are tangent in consecutive pairs, then the lines AD, BE, CF meet in a common point.
A proof via Laguerre transformations is essentially trivial. Sorry about that, Ed! --- WFL
Well, that statement was fairly garbled --- but it hardly matters, since I now have to withdraw the withdrawal --- sadly, it ain't so! It eventually dawned on me that the configuration of eight circles admits a more symmetric formulation: Take four arbitrary (oriented) circles P1,P2,P3,P4 [ex A,C,E,Q]; construct (choosing one of each pair possible) the circles Q1,Q2,Q3,Q4 [ex B,D,F,P] tangent to (P2,P3,P4), (P3,P4,P1), (P4,P1,P2), (P1,P2,P3) resp; join by lines the points of tangency of one opposing pair --- e.g. (Q4,P4) --- with (P1,Q1), (P2,Q2), (P3,Q3) resp. These lines are claimed to be concurrent, in a point R4; furthermore by symmetry, there should be in total four such points R1,R2,R3,R4. Consideration of a simple bilaterally symmetric example having just two such points establishes that this unrestricted claim is fallacious. A deceptively convincing diagram seems to have been an instance of its author's inadvertant capacity to generate apparently random examples that support unjustifiable hypotheses. I trust nobody spent too much time attempting to reconstruct the allegedly trivial proof. However, a special case for which the claim does hold true is a triangle together with its circum- and in-circle: the joins from vertices to in-circle tangencies meet the sides at intercepts s-a, s-b, s-c twice over, hence are are concurrent by Ceva's theorem. Six of the circles have zero or infinite radii; and the other three concurrences at the vertices are trivial. This instance, along with the Seven Circle Theorem, suggests that there may yet be lurking an elegant common generalisation, subject to some further constraint: what might this look like? Fred Lunnon (Dept. of Red Faces)