Mediatrix of two arcs of a quasicircle (Q1174678)

Given two nonempty disjoint closed subsets \(E\) and \(F\) of the plane, consider the set \(M\) of points in the plane at equal distance to \(E\) and \(F\). For example, if \(E\) and \(F\) are intervals on a line \(L\), then \(M\) is a line perpendicular to \(L\). It is natural to hope that \(M\) should have some special structure even for fairly general sets \(E\) and \(F\), and that \(M\) might be qualitatively similar to the simple example above under reasonable hypotheses on \(E\) and \(F\). Some results of this type are obtained in this paper, and in particular it is shown that \(M\) must be a chord-arc curve if \(E\) and \(F\) are two unbounded arcs of a quasicircle. One can ask analogous questions in higher dimensions, but one won't find any answers in this paper.