In [l] Melzak has posed the following problem: “A plane finite set Xn consists of n ≥ 3 points and contains together with any two points a third one, equidistant from them. Does Xn exist for every n ? Must it consist of points lying on some two concentric circles (one of which may reduce to a point)? How many distinct (that is, not similar) Xn are there for a given n ? …” We shall here provide a construction for uncountably many Xn for every n > 4, and a counterexample to the second question above.