Vertex positions of the generalized orthocenter and a related elliptic curve

TL;DR

This paper explores the geometric loci related to the generalized orthocenter in triangles, revealing that certain special points form elliptic curves and providing synthetic constructions for these loci.

Contribution

It characterizes the set of points where the generalized orthocenter coincides with a vertex as a union of ellipses minus six points, and studies the elliptic curve formed by points where a specific affine map is a translation.

Findings

The locus of points P with H coinciding with a vertex is a union of three ellipses minus six points.

The set of points P where a certain affine map is a translation forms an elliptic curve minus six points.

Synthetic geometric constructions for these loci are provided.

Abstract

We study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex $A,B$, or $C$. The set of all such points $P$ is a union of three ellipses minus $6$ points. In addition, if $T_P$ is the affine map taking $ABC$ to the cevian triangle $DEF$ of $P$ with respect to $ABC$, $P'$ is the isotomic conjugate of $P$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle of $P'$, then we study the locus of points $P$ for which the map $\textsf{M}_P=T_p \circ K^{-1} \circ T_{P'}$ is a translation. Here, $K$ is the complement map for $ABC$, and $\textsf{M}_P$ is an affine map taking the circumconic of $ABC$ for $P$ to the inconic of $ABC$ for $P$. The locus in question turns out to be an elliptic curve minus $6$ points, which can be synthetically constructed using the geometry of the triangle.