[math-fun] Yet another irritating elementary geometry puzzle

I finally wrote up properly all the stuff I've been doing about inspheres and exspheres of a simplex. Finished. Done & dusted. Off my plate. But as I sat down again to start writing up P*nc*l*t's perishin' p*r*sm, an obnoxious inspiration intruded. Why shouldn't a simplex in Euclidean space also sport an ` m-fold' exsphere, tangent to m facets on their exterior side, and to n-m+1 on their interior side, where perhaps m > 1 ? In particular, show (A) That m = 2 is impossible when n = 2 (for a triangle); (B) That m = 2 is impossible when n = 3 (for a tetrahedron); (C) What happens for m = 2 and n = 4 (for a pentatope); (D) What happens for general 0 <= m <= n+1 ? Fred Lunnon