Around July 1997 under the subject line "geometry puzzle", there was a discussion in math-fun about Ptolemy's and Purser's theorems, which I did my best to derail by whacking Ptolemy into n-space using determinants. I recently came across this thread again, recalled failing to discover what Purser's theorem said, and while repairing the omission encountered Casey's theorem: [See http://mathworld.wolfram.com/CaseysTheorem.html; PursersTheorem.html] Four (oriented) circles in the Euclidean plane are tangent to a fifth (circle or line) just when t_12 t_34 +/- t_13 t_24 +/- t_14 t_23 = 0, where t_ij denotes the length of tangent common to i-th and j-th circles. For this to be interpreted correctly circles and lines must be oriented: think of them as being hairy on one side or the other, the hairyness matching at points of tangency. [Synthetic proofs not being my forte, I leave them to those better equipped.] Casey's theorem follows in a way similar to Ptolemy's, from the case n = 2 of another beautiful determinant which I didn't mention previously: Given n+4 oriented spheres in Euclidean n-space, the determinant of their common tangent squared lengths has value zero: | (t_ij)^2 | = 0. There is a puzzling feature of Casey's theorem which seems to have gone unremarked. There are two distinct ways in which a fifth (oriented) circle may be tangent to four others: e.g. in the simplest case where all four are external to each other and oriented alike, a tangent circle may be either surrounded by them or surrounding them; but the Casey criterion fails to distinguish these alternatives. How may the two cases be distinguished? More interestingly, what is an analogous criterion for four circles to be tangent to two others? Fred Lunnon 23/12/05