A cevian locus and the geometric construction of a special elliptic curve

Abstract

In a previous paper we defined the circumconic of a triangle $ABC$ with respect to a point $P$ as the conic $\tilde{C}=T_{P^{\prime }}^{-1}(N_{P^{\prime }})$, where $N_{P^{\prime }}$ is the $9$-point conic for the quadrangle $ABC{P}^{\prime }$ with respect to the line at infinity, $P^{\prime }$ is the isotomic conjugate of $P$ with respect to $ABC$, and $T_{P^{\prime }}$ is the affine map taking $ABC$ to the cevian triangle for $P^{\prime }$. In this paper we determine the locus of points for which a certain affine map $\text{M}$ taking the circumconic $\tilde{C}$ to the inconic $\mathcal{I}$, defined to be the unique conic tangent to the sides of $ABC$ at the traces of the point $P$ on those sides, is a half-turn. This locus turns out to be an elliptic curve minus six points, which can be constructed geometrically using a family of affine maps defined for points on three open arcs of a circle.

arXiv comment18 pages, 4 figures

References in corpus (3)

• Synthetic foundations of cevian geometry II: The center of the cevian conic
• Vertex positions of the generalized orthocenter and a related elliptic curve
• Synthetic foundations of cevian geometry, IV: the TCC-perspector theorem