We introduce and study a geometric modification of the Douglas– Rachford 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.