
Journal for Geometry and Graphics 21 (2017), No. 1, 007027 Copyright Heldermann Verlag 2017 Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve Igor Minevich Dept. of Mathematics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 024673806, U.S.A. igor.minevich@bc.edu Patrick Morton Dept. of Mathematical Sciences, Indiana & Purdue University, 402 N. Blackford Street, Indianapolis, IN 46202, U.S.A. pmorton@math.iupui.edu [Abstractpdf] 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. Keywords: Generalized orthocenter, circumconic, inconic, affine maps, elliptic curve. MSC: 51A05; 51A20, 51M99, 51N10 [ Fulltextpdf (859 KB)] 