ISOSCELES PLANAR SUBSETS

A finite planar set is k-isosceles for k greater than or equal to 3 if every k-point subset of the set contains a point equidistant from two others. There are three nonsimilar 3-isosceles sets with five points and one with six points. Eleven 4-isosceles sets with eight points are noted, and it is conjectured that no 4-isosceles set has nine points. Exactly one 4-isosceles 8-set has four points on a line, and every 4- isosceles set that includes the vertices of a square has fewer than ei ght points.