On the bisection method for triangles

Abstract

Let <italic>UVW</italic> be a triangle with vertices <italic>U, V</italic> , and <italic>W</italic> . It is "bisected" as follows: choose a longest edge (say <italic>VW</italic> ) of <italic>UVW</italic> , and let <italic>A</italic> be the midpoint of <italic>VW</italic> . The <italic>UVW</italic> gives birth to two daughter triangles <italic>UVA</italic> and <italic>UWA</italic> . Continue this bisection process forever. We prove that the infinite family of triangles so obtained falls into finitely many similarity classes, and we obtain sharp estimates for the longest <italic>j</italic> th generation edge.