On point sets containing their triangle centers

Let S be a set of points in the plane, not all on a line, such that for every three noncollinearpoints in S, the incenter of the triangle determined by these three points is also in S. We show that S, the closure of S, is the convex hull of S. When the incenter is replaced with the circumcenter inthe previous statement, we prove that S is everywhere dense in the plane; that is S = R2. Theseresults resolve open problems posed by G. Martin in [M05]. Furthermore, we discuss similar problems involving other major triangle centers and propose the study of several other iterativeprocedures in the plane.