Diophantine equation related to angle bisectors and solutions of Pell's equations

TL;DR

This paper explores a specific Diophantine equation related to rational slopes of lines and angle bisectors, linking solutions to Pell's equations and Pythagorean triples, with implications for geometric drawing techniques.

Contribution

It provides a complete characterization of integral solutions to the Diophantine equation using properties of Pell's equations and introduces formulas for rational solutions via Pythagorean triples.

Findings

All nontrivial integral solutions are described, including those involving negative Pell's equations.

The solutions are connected to properties of Pell numbers and half-companion Pell numbers.

A formula for rational solutions derived from Pythagorean triples is presented.

Abstract

It is important in drawing techniques to find combinations of two straight lines and their angle bisectors whose slopes are all rational numbers. This problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ In this article, we describe all nontrivial integral solutions of the equation with solutions of negative Pell's equations. The formula is proven by certain properties of solutions of Pell's equations like those of half-companion Pell numbers and Pell numbers. We also give a formula for its rational solutions produced by Pythagorean triples with identical legs.