Rational quadrilaterals

TL;DR

This paper introduces new, simpler parametrizations for generating all rational quadrilaterals, including cyclic and noncyclic types, expanding on historical methods and providing a comprehensive framework.

Contribution

It presents an alternative, simplified method for parametrizing all rational quadrilaterals, including complete parametrizations for cyclic cases and new parametrizations for noncyclic cases.

Findings

Complete parametrization of rational cyclic quadrilaterals

Simpler parametrizations for noncyclic rational quadrilaterals

Framework for deriving further parametrizations

Abstract

A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an elegant solution of the problem. In 1848 Kummer gave a method of generating all rational quadrilaterals. In this paper we present an alternative method of generating all rational quadrilaterals. For rational cyclic quadrilaterals, we obtain a complete parametrization and for noncyclic rational quadrilaterals, we give several parametrizations in terms of quadratic and quartic polynomials. The parametrizations obtained in this paper are simpler than the known parametrizations of rational quadrilaterals. We also describe how further parametrizations of rational quadrilaterals may be obtained.