  For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one
defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and
$q$. It is known that mediatrices have a cell decomposition consisting of a
finite number of branch points connected by Lipschitz curves. This paper
establishes additional geometric regularity properties of mediatrices. We show
that mediatrices have the radial linearizability property, which implies that
at each point they have a geometrically defined derivative in the branching
directions. Also, we study the particular case of mediatrices on spheres, by
showing that they are Lipschitz simple closed curves exhibiting at most
countably many singularities, with finite total angular deficiency.
