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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.

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