On a series of centers of any given triangle

Given any triangle T and any positive integer n >=3, we can define a center C_n of T that is the center of gravity of the largest regular n-gon that is contained in T. For large n, C_n tends to the incenter of T.

Question:If we connect these C_n's with a smooth curve, what will be the curve on which all C_n's lie? Is it a spiral converging to the incenter?

Note: One can think of a similar series of centers of smallest regular n_gons that contain T - and this set might give a more interesting smooth curve when joined.