Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve

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