More about 4-isosceles planar sets

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 the other two. In [1] Fishburn obtains several important results about isosceles planar sets and poses a series of conjectures and open questions. We disprove Conjecture 1 in [1] and provide another 34 nonsimilar 4-isosceles 8-point planar sets which answer one of the open questions in [1] negatively.