  Phylogenetic networks are generalizations of trees that allow for the
modeling of non-tree like evolutionary processes. Split networks give a useful
way to construct networks with intuitive distance structures induced from the
associated split graph. We explore the polyhedral geometry of distance matrices
built from circular split systems which have the added property of being
equidistant. We give a characterization of the facet defining inequalities and
the extreme rays of the cone of distances that arises from an equidistant
network associated to any circular split network. We also explain a connection
to the Chan-Robbins-Yuen polytope from geometric combinatorics.
