Cauchy theorem for a triangle (${\oint }_{\partial T}f(z)dz=0$)

A holomorphic function on an open neighbourhood of a closed triangle has zero contour integral around its boundary. Goursat's subdivision proof uses complex differentiability alone: repeatedly choose a quarter-size triangle carrying at least one quarter of the integral, and subtract the affine expansion at the limiting point. No continuity of the derivative is assumed.