Rational Angle Bisection Problem in Higher Dimensional Spaces and Incenters of Simplices over Fields

TL;DR

This paper generalizes the rational angle bisection problem to higher dimensions over subfields of real numbers, providing characterizations, formulas, and conditions for rationality of incenters in simplices.

Contribution

It introduces new characterizations and formulas for rational angle bisectors and incenters of simplices over fields, extending classical geometric problems to higher dimensions.

Findings

Characterizations of when angle bisectors have rational direction vectors.

A formula for solutions to a generalized negative Pell's equation.

Necessary and sufficient conditions for the incenter of a simplex to be rational.

Abstract

In this article, we generalize the following problem, which is called the rational angle bisection problem, to the $n$-dimensional space $k^n$ over a subfield $k$ of $\mathbb R$: in the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between the two lines with slopes $a$ and $b$ rational? First, we provide several characterizations of when the angle bisectors between two lines with direction vectors in $k^n$ have direction vectors in $k^n.$ To find solutions to the problem in the case when $k = \mathbb Q,$ we derive a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we establish a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. In the coordinate plane, we prove that every triangle with $k$-rational vertices and incenter can be obtained by scaling a triangle with $k$-rational side lengths and area, which is a generalization of a Heronian triangle. We also discuss certain fundamental properties of a few centers of a given triangle with $k$-rational vertices.