Apropos this problem, suppose we're given any triangle T and its incircle C. C is tangent at 3 points of T = T_1, which we take as vertices of a new triangle T_2. Etc. If this procedure is repeated infinitely many times, the T_k's seem to converge to a single point P = P(T). Is this a well-known "triangle center", and if so what is it called and what are some of its known properties? Assuming this sequence always converges to a point, then in the unlikely event no one has named it yet, I propose calling it the "ultracenter" of T. --Dan