Pivot-isogonal cubics (Q286682)

Given a fixed point \(V\) and an involutory quadratic transformation \(\phi\) of the real projective plane, the locus of points \(P\) such that \(P\), \(\phi(P)\) and \(V\) are collinear is a cubic curve \(J\) through \(V\) which is invariant under \(\phi\). An example of such a transformation is the isogonal transform with respect to a triangle. Here, the polar conic \(K\) of \(J\) at \(V\) is a right hyperbola containing the triangle's in- and excenters [\textit{H. M. Cundy} and \textit{C. F. Parry}, J. Geom. 53, No. 1--2, 41--66 (1995; Zbl 0889.51030)]. As \(V\) varies on \(J\), the polar conics vary in a pencil of right hyperbolas. The main result of this paper is a certain converse and generalization of above theorem. The isogonal transform is replaced by a Cremona transform with four (possibly complex and/or coinciding) fixpoints. They give rise to a cubic construction and an associated pencils of conics which consists of right hyperbolas if and only if the Cremona transformation is of a generalized isogonal type. The classical result (and its converse) is recovered if the pencil of right hyperbolas has four real base points. Throughout the paper, the authors pay attention to avoid complex coordinates by considering a pencil of conics instead of points of a quadrilateral. They also briefly mention possible extensions to cubic constructions via other triangle centers but do not go into detail.