  Motivated by the circumcentered Douglas--Rachford method recently introduced
by Behling, Bello Cruz and Santos to accelerate the Douglas--Rachford method,
we study the properness of the circumcenter mapping and the circumcenter method
induced by isometries. Applying the demiclosedness principle for circumcenter
mappings, we present weak convergence results for circumcentered isometry
methods, which include the Douglas--Rachford method (DRM) and circumcentered
reflection methods as special instances. We provide sufficient conditions for
the linear convergence of circumcentered isometry/reflection methods. We
explore the convergence rate of circumcentered reflection methods by
considering the required number of iterations and as well as run time as our
performance measures. Performance profiles on circumcentered reflection
methods, DRM and method of alternating projections for finding the best
approximation to the intersection of linear subspaces are presented.
