  In this paper, it is shown that a Fuchsian group, acting on the upper
half-plane model for $\mathbb{H}^2$, admits a Ford domain which is also a
Dirichlet domain, for some center, if and only if it is an index 2 subgroup of
a reflection group. This is used to exhibit an example of a maximal arithmetic
hyperbolic reflection group which is not congruence. Analogous results, and
counterexamples, are given in the case of Kleinian groups.
