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

TL;DR

This paper explores a special elliptic curve arising from a geometric locus related to cevian configurations and affine maps in a triangle, providing a new geometric construction of the curve.

Contribution

It characterizes the locus of points where a specific affine map is a half-turn as an elliptic curve minus six points, with a geometric construction method.

Findings

The locus is an elliptic curve minus six points.

A geometric construction of the elliptic curve using affine maps and circle arcs.

Identification of conditions for the affine map to be a half-turn.

Abstract

In a previous paper we defined the circumconic of a triangle $ABC$ with respect to a point $P$ as the conic $\tilde C=T_{P'}^{-1}(N_{P'})$, where $N_{P'}$ is the $9$-point conic for the quadrangle $ABCP'$ with respect to the line at infinity, $P'$ is the isotomic conjugate of $P$ with respect to $ABC$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle for $P'$. In this paper we determine the locus of points for which a certain affine map $\textsf{M}$ 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.