  Consider a geodesic triangle on a surface of constant curvature and subdivide
it recursively into 4 triangles by joining the midpoints of its edges. We show
the existence of a uniform $\delta>0$ such that, at any step of the
subdivision, all the triangle angles lie in the interval $(\delta, \pi
-\delta)$. Additionally, we exhibit stabilising behaviours for both angles and
lengths as this subdivision progresses.
