Dan Asimov dasimov at earthlink.net Just noticed this problem, with one solution given, in an obscure article: "Find a finite subset of the plane such that for any two of its points the perpendicular >bisector of the segment connecting them passes through exactly two points of the set."

--don't know answer, but if we transplant to surface of sphere not plane (use geodesics not lines) then the 4 vertices of a regular tetrahedron work. (Other platonics have other interesting properties of this ilk.)