Approximation of Quadrilaterals by Rational Quadrilaterals in the Plane

TL;DR

This paper investigates the density of rational quadrilaterals and triangles in the plane by transforming geometric problems into Diophantine equations on elliptic curves, providing new proofs and criteria for density in various geometric contexts.

Contribution

It introduces a novel approach linking geometric approximation problems to elliptic curve theory, establishing density results for rational quadrilaterals and triangles, and develops a new criterion for density in topological spaces.

Findings

Density of rational quadrilaterals in all quadrilaterals established

Density of rational triangles related to rational points on the unit circle

Parallelograms with rational sides and area are dense in all parallelograms

Abstract

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides,diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transformed to a two parameter family of elliptic curves to deduce our main result concerning density of points on a line which are at a rational distance from three collinear points (Theorem 4). We deduce from this a new proof of density of rational quadrilaterals in the space of all quadrilaterals (Theorem 39). The other main result (Theorem 3) of this article is on the density of rational triangles which is related to analyzing rational points on the unit circle. Interestingly, this enables us to deduce that parallelograms with rational sides and area are dense in the class of all parallelograms. We also give a criterion for density of certain sets in topological spaces using local product structure and prove the density Theorem 6 in the appendix section. An application of this proves the density of rational points as stated in Theorem 31.