Real elliptic curves and cevian geometry

TL;DR

This paper introduces a geometric normal form for real elliptic curves and links their points to cevian geometry of triangles, providing a new geometric perspective on elliptic curves with real j-invariant.

Contribution

It establishes that all elliptic curves with real j-invariant are isomorphic to a specific geometric normal form and characterizes their points via cevian geometry.

Findings

Elliptic curves with real j-invariant can be represented in a geometric normal form.

Points on these curves, minus six exceptions, relate to cevian geometry.

The geometric normal form simplifies understanding of real elliptic curves.

Abstract

We study the elliptic curve $E_a: (ax+1)y^2+(ax+1)(x-1)y+x^2-x=0$, which we call the geometric normal form of an elliptic curve. We show that any elliptic curve whose $j$-invariant is real is isomorphic to a curve $E_a$ in geometric normal form, and show that for $a \notin \{0, -1, -9\}$, the points on $E_a$, minus a set of $6$ points, can be characterized in terms of the cevian geometry of a triangle.