On Nested Sequences of Triangles

Abstract

For a triangle \({\mathbb{T}}\) and a real number s, the s-medial triangle \({\mathcal{M}}_{s}({\mathbb{T}})\) of \({\mathbb{T}}\) is the triangle whose vertices divide the sides of \({\mathbb{T}}\) in the ratio s : 1 − s. In this paper, we answer a question raised in [8] regarding the values of s for which the sequence \({\mathcal{M}}_s^n({\mathbb{T}}), n = 1, 2, \ldots ,\) of triangles converges in shape, and we prove that if \({\mathbb{T}}\) is not equilateral, then s = 1/2 is the only such value. We also characterize those values of s for which the sequence is periodic and those values for which it is everywhere dense. We do the same for two other related sequences of triangles. As a side-product, we obtain a simple proof, and extensions, of a theorem of A. Emmerich regarding triangles with equal Brocard angles.