New results on the dynamic geometry generated by sequences of nested triangles

Abstract

Starting from an initial triangle, one may wish to check whether a sequence of iterations is convergent, or is convergent in some shape, and to find the limit. In this paper we first prove a general result for the convergence of a sequence of nested triangles (Theorem 2.2), then we study some properties of the power curve $\Gamma$ of a triangle. These are used to prove that the sequence of nested triangles defined by a point $Q^{(s)}$ on the power curve converges to a point for every $s\in [0,2]$ (Theorem 4.2). In particular, we obtain that the sequence of nested triangles defined by the incenter converges to a point, completing the main result in Ismailescu, D.; Jacobs, J. On sequences of nested triangles. {\em Period. Math. Hungar.} {\bf 53} (2006), no.(1-2), 169--184. Finally, we present some numerical simulations which inspire open questions regarding the convergence of such iterations.