Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve

I. Minevich ; P. Morton

Journal for geometry and graphics, Tome 21 (2017) no. 1, pp. 7-27

Résumé

We study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex. The set of all such points $P$ is a union of three ellipses minus six points. If $T_P$ is the affine map taking $ABC$ to the cevian triangle of $P$, $P'$ is the isotomic conjugate of $P$, and $K$ is the complement map for $ABC$, we also study the affine map $M_P=T_P \circ K^{-1} \circ T_{P'}$ taking the circumconic of $ABC$ for $P$ to the inconic of $ABC$ for $P$. We show that the locus of points $P$ for which this map is a translation is an elliptic curve minus six points, and show how this locus can be synthetically constructed using the geometry of the triangle.

Détail

Classification : 51A05, 51A20, 51M99, 51N10
Mots-clés : Generalized orthocenter, circumconic, inconic, affine maps, elliptic curve