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On Finite Plane Sets Containing for Every Pair of Points an Equidistant Point
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1967
References
1.
Melzak, Z. A., Problems connected with convexity. Canad. Math. Bull. 8 (1966), pages 565-573: Problem (21).CrossRefGoogle Scholar
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