Our original results refer to dualistic structure on primaldual interior-point methods for symmetric cone programs with linear constraints. It is shown that scalings by the Nesterov-Todd direction are generated by middle points of geodesics joining with primal interior points and dual interior points. Finally we relate power classes of search directions with geodesics and weighted geometric means. M.S.C. 2010: 17C36, 53A15, 90C51.