Geometrical properties of some Euler and circular cubics. I (Q1818901)

For any point \(F\) on the Euler line of a triangle, the locus of the points \(P\) such that it, its isogonal conjugate, and \(F\) are collinear is a cubic curve, one of the pencil of ``Euler cubics'' of the triangle. These have been studied since the 19th century, and are today somewhat obscure; however, as the authors show, there is still something new to add. This paper, which investigates the properties of these, is a sequel to the authors' 1995 paper [J. Geom. 53, No. 1-2, 41-66 (1995; Zbl 0889.51030)]. It summarizes some older results, and presents several new ones (most of which are difficult to state without first introducing rather a lot of notation). The illustrations are, fortunately, plentiful.