, Volume 66, Issue 1-2, pp 55-71

On the cubic of Napoleon

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Abstract

This paper explores properties of an interesting cubic from the Euler pencil of analagmatic cubics associated to a triangle. Our arguments are algebraic and use a combination of trilinear coordinates and complex numbers. We study its basic properties with emphasis on ways how to recognise this curve in various geometric configurations. Since this cubic is closely tied with the so called Napoleon configuration of a scalene triangle, we named it the Napoleon cubic.