NO CONTINUUM IN E2 HAS THE TMP .2. TRIODIC CONTINUA

A subset X of the Euclidean plane E2 has the triple midset property (TMP) if, for every line segment [x, y] in E2 such that {x, y} subset-of X, the perpendicular bisector of [x, y] meets X in exactly three points. Resolving the planar aspect of a more general question, the main theorem shows that no compact, connected, nondegenerate subset of E2 can possess this triple midset property.