Geodesic
On Finite Plane Sets Containing for Every Pair of Points an Equidistant Point
Canadian mathematical bulletin, Tome 10 (1967) no. 1, pp. 119-120

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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. 
Brown, W.G. On Finite Plane Sets Containing for Every Pair of Points an Equidistant Point. Canadian mathematical bulletin, Tome 10 (1967) no. 1, pp. 119-120. doi: 10.4153/CMB-1967-014-0
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     title = {On {Finite} {Plane} {Sets} {Containing} for {Every} {Pair} of {Points} an {Equidistant} {Point}},
     journal = {Canadian mathematical bulletin},
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     year = {1967},
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     doi = {10.4153/CMB-1967-014-0},
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