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

Takashi Hirotsu

TL;DR

This work extends the rational angle bisection problem to higher dimensions over a subfield $k$ of $\mathbb{R}$, linking geometric angle-bisector conditions to algebraic criteria and Pell-like equations. It develops a unified framework: (i) a set of equivalent conditions for when angle bisectors lie in $k^n$, (ii) a Pell-like Diophantine structure governing integral solutions and their geometric consequences, and (iii) criteria for the incenter of $n$-simplices and various triangle centers to be $k$-rational, including constructive procedures that relate arbitrary $k$-rational triangles to Heronian ones. The paper also demonstrates that many classical triangle centers remain $k$-rational in this setting and provides explicit constructions to realize triangles and simplices with $k$-rational centers. These results have potential implications for exact geometric modeling and computational geometry in settings where number-theoretic precision is required.

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 $\mathbb R$: on the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between two lines with slopes $a$ and $b$ rational? First, we give a few 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 also give a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of the negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. On 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 state certain fundamental properties of a few centers of a given triangle with $k$-rational vertices.

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

TL;DR

This work extends the rational angle bisection problem to higher dimensions over a subfield of , linking geometric angle-bisector conditions to algebraic criteria and Pell-like equations. It develops a unified framework: (i) a set of equivalent conditions for when angle bisectors lie in , (ii) a Pell-like Diophantine structure governing integral solutions and their geometric consequences, and (iii) criteria for the incenter of -simplices and various triangle centers to be -rational, including constructive procedures that relate arbitrary -rational triangles to Heronian ones. The paper also demonstrates that many classical triangle centers remain -rational in this setting and provides explicit constructions to realize triangles and simplices with -rational centers. These results have potential implications for exact geometric modeling and computational geometry in settings where number-theoretic precision is required.

Abstract

In this article, we generalize the following problem, which is called the rational angle bisection problem, to the -dimensional space over a subfield of : on the coordinate plane, for which rational numbers and are the slopes of the angle bisectors between two lines with slopes and rational? First, we give a few characterizations of when the angle bisectors between two lines with direction vectors in have direction vectors in To find solutions to the problem in the case when we also give a formula for the integral solutions of which is a generalization of the negative Pell's equation where is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given -simplex with -rational vertices to be -rational. On the coordinate plane, we prove that every triangle with -rational vertices and incenter can be obtained by scaling a triangle with -rational side lengths and area, which is a generalization of a Heronian triangle. We also state certain fundamental properties of a few centers of a given triangle with -rational vertices.
Paper Structure (5 sections, 14 theorems, 38 equations, 1 figure)

This paper contains 5 sections, 14 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Angle Bisectors
  3. Pythagorean and Pell-like Equations
  4. Incenters of Simplices over Fields
  5. Centers of Triangles over Fields

Key Result

Theorem 1

Let $\bm{a}$ and $\bm{b}$ be linearly independent vectors of $k^n.$

Figures (1)

  • Figure 2: The centroid $G,$ the circumcenter $E,$ the orthocenter $H,$ and the incenter $I$ of triangle $ABC$ with $A = (0,0),$$B = (17,7),$ and $C = (3,21).$

Theorems & Definitions (44)

  • Remark 1
  • Theorem 1
  • Example 1
  • Example 2
  • Definition 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm-ang']}
  • Theorem 4: Car15
  • ...and 34 more