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Countless Simson Line Configurations

C J Bradley, J T Bradley, Countless Simson Line Configurations. The Mathematical Gazette, 80 (488). ISSN 0025-5572, pp. 314–321. July 1996. PDF, 106 Kbytes.


The Simson line property is normally associated with points on the circumcircle of a triangle. It is embodied by the following theorem. Given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC, CA, AB meet those sides at L, M, N respectively then L, M, N are collinear if and only if P lies on the circumcircle of triangle ABC. The line LMN is then known as the Simson line of P. It is the word perpendicular that gives the impression that this type of configuration is somehow particular and that the Simson line property is not therefore capable of generalization. In this article we show that this is not the case and we demonstrate that the above theorem is simply one case of a more general theorem. Indeed it turns out that every transversal of a triangle is a Simson line in a more general sense, and we show how to associate these transversals with different configurations.

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