The following problem (or rather its solution) constitutes the crux of what appears to be a new result in elementary 3-space Euclidean geometry. However its solution proves so neat that I'm posing it on the list, ahead of the solution to follow in a few days. [A handy excuse for further procrastination over the next instalment of Poncelet pontification.] Gloss (0) : Given an arbitrary tetrahedron A , denote by B the inscribed tetrahedron with vertices the centroids of faces of A . It is well known that B is similar to A , but scaled down to 1/3 ; and an analogous result holds for any dimension n , with scale-down 1/n . Given A and a chosen vertex A_0 , a tetrahedron is `exscribed' when its vertices meet the extended face planes of A , but lie outside face plane F_0 opposite A_0 . Problem (1) : Construct explicitly some exscribed C congruent to A . For example, grey A and yellow C in diagram https://www.dropbox.com/s/blcihoao8t0k3lh/exscribe2.png Problem (2) : Generalise the result to a simplex in Euclidean n-space, for n >= 3 . Question (3) : Investigate whether C is the smallest possible exscribed on F_0 and similar to A . (Probably, but I don't know either.)