Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ >0 such that, at any step of the subdivision, all the triangle angles lie in the interval (δ ,π -δ ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.