Based on computing spherical lengths of polygonal curves and spherical areas of domains bounded by
polygonal curves on the complex plane, this paper proves a property of holomorphic functions and, as
an application, gives a very brief proof of a famous inequality obtained by L. V. Ahlfors which easily
implies Picard's theorem. Taken together with the arguments given by Alhfors, therefore, this paper in
fact gives an elementary, brief and geometrical proof of Picard's theorem.