Curves, Domains and Picard's Theorem

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