We study the anyon statistics of 2+1 dimensional gauge theories based on the Breit Bamiltonia.n formalism for two-fermion elastic scattering. A Breit potential, the nonrelativistic limit of the scattering amplitude, is derived to lowest order of coupling constant for Maxwell, Chern-Simons, and Maswell-Chern-Simons theories respectively. An anyon phase angle is then extracted from the spin-orbit interaction in the Breit potential. We also find that a nontrivial anyon phase is induced from higher-order radiative corrections in the pure Maxwell theory.