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Extensions of Sylvester's Theorem

Published online by Cambridge University Press:  20 November 2018

Richard H. Balomenos ,
William E. Bonnice  and
Robert J. Silverman
Richard H. Balomenos
Affiliation:
University of New Hampshire
William E. Bonnice
Affiliation:
University of New Hampshire
Robert J. Silverman
Affiliation:
University of New Hampshire
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Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 1 , April 1966 , pp. 1 - 14
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Coxeter, H. S. M., Introduction to Geometry, John Wiley and Sons, Inc., 1961, 65.Google Scholar
2. Dirac, G. A., Collinearity Properties of Sets of Points, Quart. J. Math. Oxford Ser. 2(1951), 221-227.Google Scholar
3. Erdös, P., Problem No. 4065, Amer. Math. Monthly, 50 (1943), 65.Google Scholar
4. Amer. Math. Monthly, 51 (1944) 169-171.Google Scholar
5. Kelly, L. M. and Moser, W.O.J., On the Number of Ordinary Lines Determined by n Points, Canad. J. Math. 10 (1958), 210-219.Google Scholar
6. Motzkin, Th., The Lines and Planes Connecting the Points of a Finite Set, Trans. Amer. Math. Soc. 70 (1951), 451-464.Google Scholar
7. Sylvester, J.J., Mathematical Question 11851, Educational Times, 59 (1893) 98.Google Scholar