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arXiv · 1608.04614

Vertex positions of the generalized orthocenter and a related elliptic curve

Igor Minevich·Patrick Morton
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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.

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Igor Minevich, Patrick Morton. 2016-08-15. Vertex positions of the generalized orthocenter and a related elliptic curve. https://arxiv.org/abs/1608.04614

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