Geodesic
Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve
Journal for geometry and graphics, Tome 21 (2017) no. 1, pp. 7-27.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 51A05, 51A20, 51M99, 51N10
Mots-clés : Generalized orthocenter, circumconic, inconic, affine maps, elliptic curve 
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     author = {I. Minevich and P. Morton },
     title = {Vertex {Positions} of the {Generalized} {Orthocenter} and a {Related} {Elliptic} {Curve}},
     journal = {Journal for geometry and graphics},
     pages = {7--27},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/JGG_2017_21_1_JGG_2017_21_1_a1/}
}
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I. Minevich; P. Morton . Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve. Journal for geometry and graphics, Tome 21 (2017) no. 1, pp. 7-27. http://geodesic.mathdoc.fr/item/JGG_2017_21_1_JGG_2017_21_1_a1/