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Journal for Geometry and Graphics 21 (2017), No. 1, 007--027
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 02467-3806, U.S.A.

Patrick Morton
Dept. of Mathematical Sciences, Indiana & Purdue University, 402 N. Blackford Street, Indianapolis, IN 46202, U.S.A.


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

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