  We introduce and study a geometric modification of the Douglas-Rach\-ford
method called the Circumcentered-Douglas-Rachford method. This method iterates
by taking the intersection of bisectors of reflection steps for solving certain
classes of feasibility problems. The convergence analysis is established for
best approximation problems involving two (affine) subspaces and both our
theoretical and numerical results compare favorably to the original
Douglas-Rachford method. Under suitable conditions, it is shown that the linear
rate of convergence of the Circumcentered-Douglas-Rachford method is at least
the cosine of the Friedrichs angle between the (affine) subspaces, which is
known to be the sharp rate for the Douglas-Rachford method. We also present a
preliminary discussion on the Circumcentered-Douglas-Rachford method applied to
the many set case and to examples featuring non-affine convex sets.
