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

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.