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 $ABC$, $P^{\prime }$ is the isotomic conjugate of $P$, and $T_{P^{\prime }}$ is the affine map taking $ABC$ to the cevian triangle of $P^{\prime }$, then we study the locus of points $P$ for which the map ${\text{M}}_P=T_p\circ K^{-1}\circ T_{P^{\prime }}$ is a translation. Here, $K$ is the complement map for $ABC$, and ${\text{M}}_P$ 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.

arXiv comment27 pages, 8 figures. arXiv admin note: text overlap with arXiv:1506.06253

Cited by in corpus (2)

• Synthetic foundations of cevian geometry, IV: the TCC-perspector theorem
• A cevian locus and the geometric construction of a special elliptic curve