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For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq 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. In the case of a topological sphere, mediatrices are called equators and it can be noticed that there are no branching points, thus an equator is a topological circle with possibly many Lipschitz singularities. This paper establishes that mediatrices have the radial linearizability property. This is a regularity property that implies that at each singular or branching point mediatrices have a geometrically defined derivative in each direction. In the case of equators we show that there are at most countably many singular points and the sum of the angles over all singularities is always finite.

Equators have at most countable many singularities with bounded total angle