Abstract The masses of the ground-state baryons are calculated in a relativistic quark model by using the saddle-point variational method to solve the three-body Breit equation. The saddle-point variational method maximizes the energy with respect to small component parameters, while minimizing with respect to size parameters. This removes the problems usually caused by negative-energy states, without the need for positive-energy projection operators. The variational method, applied to an asymmetric trial wave function with naturally borken SU(3) and SU(6) symmetry, permits solution of the Breit equation for realistic QCD-inspired potentials without using perturbation theory. The calculated masses are in good agreement with experiment, and the asymmetric wave function gives reasonable values for the proton and neutron charge radii.