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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

W.G. Brown
W.G. Brown*
Affiliation:
University of British Columbia
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 1 , April 1967 , pp. 119 - 120
Copyright
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