On Parametric Spaces of Bicentric Quadrilaterals

TL;DR

This paper explores the geometric problem of bicentric quadrilaterals with rational sides by translating it into the study of elliptic curves, revealing infinite families of such curves with specific torsion structures and ranks.

Contribution

It formulates the problem in terms of elliptic curves, identifies infinite families with high rank and specific torsion subgroups, and provides explicit examples.

Findings

Existence of infinitely many elliptic curves with rank ≥ 2 and torsion subgroup Z/8Z.

Construction of elliptic curves with torsion subgroup Z/2Z × Z/8Z.

Explicit examples of elliptic curves with rank 5.

Abstract

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by $E_a:y^2=x^3+(a^4-4a^3-2a^2-4a+1)x^2+16a^4x$ which have, in general, \(\mathbb Z/8\mathbb Z\), and in rare cases \(\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z\) as torsion subgroups. We show the existence of infinitely many elliptic curves $E_a$ of rank at least two with torsion subgroup $\mathbb Z/8\mathbb Z$, parameterized by the points of an elliptic curve of rank at least one, and give five particular examples of rank $5$. We, also, show the existence of a subfamily of $E_a$ whose torsion subgroup is $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$.