Real elliptic curves and cevian geometry

Igor Minevich; Patrick Morton

paper

Real elliptic curves and cevian geometry

arXiv:1711.11415

Abstract

We study the elliptic curve $E_{a} : (a x + 1) y^{2} + (a x + 1) (x - 1) y + x^{2} - x = 0$ , 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 $a \in / {0, - 1, - 9}$ , the points on $E_{a}$ , minus a set of $6$ points, can be characterized in terms of the cevian geometry of a triangle.

12 pages