Recapitulating "Euler's theorem", badly garbled in an earlier post of mine: Relates circumradius R , inradius r , centre displacement d of triangle by R^2 - 2 R r - d^2 = 0 . Actually due to Chapple, according to http://mathworld.wolfram.com/EulerTriangleFormula.html . Synthetic proof at https://en.wikipedia.org/wiki/Euler's_theorem_in_geometry#/media/File:GeometryEulerTheorem.png . Provides foundation of Poncelet's porism for triangles in Euclidean 2-space; see http://mathworld.wolfram.com/PonceletsPorism.html . My Scrooge's Christmas present (to myself, naturally) was to investigate the extent to which Chapple's result generalises to higher dimensions: specifically, *** What can be established about the relation between circumradius, *** *** inradius and displacement of a tetrahedron in Euclidean 3-space? *** I'll post more about this problem after a few days, in case other social misfits out there are motivated to tackle it themselves, or alternatively can disabuse me of any delusional claim to originality. Incidentally, Poncelet's porism now has an entire book devoted to it, Vladimir Dragović, Milena Radnović "Integrable Billiards" Birkhäuser 2011 ; chapter 1 free online at http://www.springer.com/cda/content/document/cda_downloaddocument/9783034800... . Fred Lunnon