No continuum in 𝐸² has the TMP. I. Arcs and spheres

Abstract

A subset X X of the Euclidean plane E 2 {E^2} is said to have the triple midset property (TMP) if, for each pair of points x x and y y of X X , the perpendicular bisector of the segment joining x x and y y intersects X X at exactly three points. In this paper it is proved that no arc or simple closed curve in E 2 {E^2} can have the TMP. In a subsequent paper these results are used to prove that no planar continuum can have the TMP.