Original Research

A generalised Sylvester-Gallai Theorem

L. M. Pretorius, K. J. Swanepoel
Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie | Vol 26, No 1 | a118 | DOI: https://doi.org/10.4102/satnt.v26i1.118 | © 2007 L. M. Pretorius, K. J. Swanepoel | This work is licensed under CC Attribution 4.0
Submitted: 21 September 2007 | Published: 21 September 2007

About the author(s)

L. M. Pretorius, Departement Wiskunde en Toegepaste Wiskunde, Universiteit van Pretoria, Pretoria 0002, South Africa
K. J. Swanepoel, Departement Wiskundige Wetenskappe, Universiteit van Suid-Afrika, Posbus 392, UNISA, Pretoria 0003, South Africa

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Abstract

We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:

Let S be a finite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia) the colour different from k and is collinear with A and B. Then all the points in S are collinear.
This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.


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