The sequence of generalized median triangles and a new shape function

Starting with a triangle ABC and a real number s, we let AA _s , BB _s , CC _s be the cevians that divide the sides BC, CA, AB, respectively, in the ratio s : 1 ? s, and we let ${\mathcal{H}_s(ABC)}$ be the triangle whose side lengths are equal to those of AA _s , BB _s , CC _s . We investigate the sequence of (the shapes of) triangles ${\mathcal{H}_s^n(ABC)}$ , n = 1, 2, ... by introducing a new shape function that suits this sequence. We also use this shape function to prove a theorem of C. F. Parry concerning automedian triangles.