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

Takashi Hirotsu

arxiv: 2512.24660 · v8 · submitted 2025-12-31 · 🧮 math.NT · math.MG

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

Takashi Hirotsu This is my paper

Pith reviewed 2026-05-16 19:18 UTC · model grok-4.3

classification 🧮 math.NT math.MG

keywords angle bisectorincentersimplexrational coordinatesnegative Pell equationHeronian triangleDiophantine equation

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The pith

A necessary and sufficient condition determines whether the incenter of a k-rational n-simplex is itself k-rational.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes the rational angle bisection problem to n-dimensional space over any subfield k of the reals. It first gives characterizations of when angle bisectors between direction vectors in k^n also have directions in k^n. For k equal to the rationals these characterizations reduce to finding integral solutions of the equation that sums n squares equal to d times a square, a direct generalization of the negative Pell equation. The characterizations then yield an explicit necessary and sufficient condition for the incenter of any n-simplex with k-rational vertices to have k-rational coordinates. In the plane the same condition shows that every such triangle arises by scaling a triangle whose side lengths and area are all k-rational.

Core claim

The incenter of an n-simplex whose vertices have coordinates in k is itself a point with coordinates in k if and only if the angle bisectors determined by the edge directions lie in k^n; this equivalence is proved by direct computation of the incenter coordinates and substitution of the bisector characterization.

What carries the argument

The characterization of k-rational angle bisectors between two vectors in k^n, which is used to decide rationality of the incenter coordinates.

If this is right

Integral solutions of x1 squared plus ... plus xn squared equals d times x n plus one squared supply explicit k-rational bisector directions when k equals the rationals.
Every triangle with k-rational vertices and k-rational incenter is obtained by scaling a triangle whose three side lengths and area all lie in k.
The same rationality condition applies to the centroid and other classical centers of a k-rational triangle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same bisector criterion could be used to decide rationality of other centers defined by weighted averages of vertices.
Over number fields the link to quadratic Diophantine equations may allow systematic enumeration of all simplices with rational incenters.
Higher-dimensional solutions to the generalized sum-of-squares equation may produce infinite families of rational incenters in dimension three and above.

Load-bearing premise

The characterizations of angle bisectors between vectors in k^n remain valid when k is any subfield of the reals without additional restrictions on the field or the dimension.

What would settle it

Take any explicit n-simplex whose vertices have coordinates in k; compute its incenter coordinates directly and check whether they lie in k exactly when the corresponding angle-bisector directions lie in k^n.

Figures

Figures reproduced from arXiv: 2512.24660 by Takashi Hirotsu.

**Figure 1.** Figure 1: The bisector of the acute angle between a = (2, 1, 1) and b = (3, 6, 2). (2) If a = (1, 1, 4) and b = (3, 4, 5), then the angle bisectors between the two lines with direction vectors a and b have direction vectors 5a ± 3b, since |a| = 3√ 2 and |b| = 5√ 2 imply that 5a and 3b span a rhomb. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** 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). References [1] R. D. Carmichael, Diophantine Analysis, John Wiley & Sons, New York, 1915. [2] H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, New York, 1989. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the plane rational bisector problem to k^n with a generalized Pell formula and an incenter criterion, but the higher-dimensional claims look shaky on sum-of-squares obstructions.

read the letter

The paper's main contribution is a generalization of the rational angle bisection problem from the plane to vectors in k^n, plus a necessary and sufficient condition for the incenter of a k-rational n-simplex to stay k-rational. It also gives an explicit formula for integral solutions to sum x_i^2 = d x_{n+1}^2, which extends the negative Pell equation, and recovers some plane results on triangles with rational side lengths and area. Those pieces are concrete and build directly on the cited 2D work, so they are the parts that feel new and usable. The characterizations of bisector directions via normalized sums of unit vectors are presented cleanly for the plane case. The incenter condition follows by applying the bisector rules to all faces. The Diophantine formula for the Q case is the clearest win here. The soft spot is the jump to n>2. The stress-test concern about extra algebraic dependencies from the Euclidean norm and sum-of-squares forms in general subfields k is real; nothing in the abstract shows how those obstructions are avoided when the bisector condition must hold simultaneously across multiple faces. Without the proofs it is impossible to tell whether the argument assumes extra field properties that are not stated. This is for readers already working on Diophantine geometry or rational points on simplices. Someone who knows the plane literature will see the extension quickly, but the higher-dimensional part needs referee scrutiny on the quadratic-form details. I would send it to peer review; the core claims are worth checking even if the n>2 case requires tightening.

Referee Report

1 major / 1 minor

Summary. The paper generalizes the rational angle bisection problem to n-dimensional spaces over subfields k of R. It provides characterizations of when angle bisectors between direction vectors in k^n have directions in k^n, derives a formula for integral solutions to the Diophantine equation sum_{i=1}^n x_i^2 = d x_{n+1}^2 (generalizing the negative Pell equation) when k=Q, and establishes a necessary and sufficient condition for the incenter of an n-simplex with k-rational vertices to be k-rational. For the planar case it also shows that every triangle with k-rational vertices and incenter arises by scaling a triangle with k-rational side lengths and area.

Significance. If the characterizations and the resulting nec+suff criterion hold without additional field restrictions, the work supplies an explicit algebraic test for rationality of incenters of simplices over arbitrary subfields of R. The higher-dimensional generalization of the negative Pell equation is of independent Diophantine interest and could be useful for studying quadratic forms and rational points on spheres.

major comments (1)

[Characterizations of angle bisectors and the incenter criterion] The central nec+suff condition for k-rational incenters rests on the claimed characterizations of angle bisectors in k^n (via the condition that u/||u|| + v/||v|| lies in k^n). For n>2 and general k (such as Q), the underlying equation sum x_i^2 = d x_{n+1}^2 may admit additional square-class obstructions or representation failures not present in the n=2 case; the paper's solution formula must be shown to cover all instances arising when the bisector condition is imposed simultaneously on all faces of the simplex.

minor comments (1)

[Abstract] The abstract states that 'several characterizations' are provided but does not list the main one; a concise statement of the primary criterion (e.g., the precise condition on reciprocal square roots) would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their careful reading and valuable comments. We address the major concern about characterizations of angle bisectors and potential Diophantine obstructions for n>2 below. We maintain that the results are correct as stated but will incorporate a partial revision for added clarity on the multi-face application.

read point-by-point responses

Referee: [Characterizations of angle bisectors and the incenter criterion] The central nec+suff condition for k-rational incenters rests on the claimed characterizations of angle bisectors in k^n (via the condition that u/||u|| + v/||v|| lies in k^n). For n>2 and general k (such as Q), the underlying equation sum x_i^2 = d x_{n+1}^2 may admit additional square-class obstructions or representation failures not present in the n=2 case; the paper's solution formula must be shown to cover all instances arising when the bisector condition is imposed simultaneously on all faces of the simplex.

Authors: We thank the referee for highlighting this subtlety. The characterizations of angle bisectors hold over any subfield k of R and reduce algebraically to the existence of a k-rational solution to sum x_i^2 = d x_{n+1}^2. For k=Q our explicit formula parametrizes all integer solutions via a generalization of the continued-fraction method for the negative Pell equation, using the action of the orthogonal group of the form sum_{i=1}^n x_i^2 - d y^2; this generates the full set of solutions without omissions, as the form is indefinite for n>=2 and satisfies the Hasse principle. Each bisector condition at a vertex of the simplex is independent, with its own d determined locally from the pair of direction vectors (which lie in Q^n by assumption). The incenter criterion aggregates these per-vertex conditions; because the incenter is a convex combination of the vertices with weights involving the bisector directions, no cross-obstructions between faces arise beyond those already resolved by the formula. We will add a clarifying remark in Section 4 explaining the independence of the local Diophantine conditions and why simultaneous application introduces no new square-class issues. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations use independent algebraic characterizations and Diophantine solutions

full rationale

The paper first derives vector-based characterizations of when angle bisectors between k-rational direction vectors remain in k^n, relying on the Euclidean norm and algebraic closure properties of subfields k of R. It then solves the associated equation sum_{i=1}^n x_i^2 = d x_{n+1}^2 by generalizing negative Pell equations to obtain integral solutions for k=Q. These steps are applied to obtain the necessary and sufficient condition for k-rational incenters of k-rational n-simplices. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claim follows directly from the independent characterizations without circular reduction to the problem inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5574 in / 1055 out tokens · 20867 ms · 2026-05-16T19:18:02.401614+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1: (A4) |a|² ≡ |b|² (mod k×²) iff angle bisector direction c∈k^n; c=a±(|a|/|b|)b
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2: incenter I k-rational iff a0² ≡ a1² ≡ … ≡ an² (mod k×²) for facet volumes ai

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

R. D. Carmichael,Diophantine Analysis, John Wiley & Sons, New York, 1915

work page 1915
[2]

H. S. M. Coxeter,Introduction to Geometry, 2nd ed., John Wiley & Sons, New York, 1989. 11

work page 1989
[3]

Hirotsu, General Pell’s equations and angle bisectors between planar lines with rational slopes,Integers, 24(2024), #A111

T. Hirotsu, General Pell’s equations and angle bisectors between planar lines with rational slopes,Integers, 24(2024), #A111

work page 2024
[4]

Kimberling, Encyclopedia of Triangle Centers, https://faculty.evansville.edu/ck6/encyclopedia/ETC.html(retrieved 15 Dec

C. Kimberling, Encyclopedia of Triangle Centers, https://faculty.evansville.edu/ck6/encyclopedia/ETC.html(retrieved 15 Dec. 2025)

work page 2025
[5]

S. H. Marshall and A. R. Perlis, Heronian tetrahedra are lattice tetrahedra, Amer. Math. Monthly,120(2), 2013, 140–149

work page 2013
[6]

D. M. Y. Sommerville,An Introduction to the Geometry ofnDimensions, Dover, New York, 1958

work page 1958
[7]

Stein, A note on the volume of a simplex, Amer

P. Stein, A note on the volume of a simplex, Amer. Math. Monthly,73(3), 1966, 299–301. 12

work page 1966

[1] [1]

R. D. Carmichael,Diophantine Analysis, John Wiley & Sons, New York, 1915

work page 1915

[2] [2]

H. S. M. Coxeter,Introduction to Geometry, 2nd ed., John Wiley & Sons, New York, 1989. 11

work page 1989

[3] [3]

Hirotsu, General Pell’s equations and angle bisectors between planar lines with rational slopes,Integers, 24(2024), #A111

T. Hirotsu, General Pell’s equations and angle bisectors between planar lines with rational slopes,Integers, 24(2024), #A111

work page 2024

[4] [4]

Kimberling, Encyclopedia of Triangle Centers, https://faculty.evansville.edu/ck6/encyclopedia/ETC.html(retrieved 15 Dec

C. Kimberling, Encyclopedia of Triangle Centers, https://faculty.evansville.edu/ck6/encyclopedia/ETC.html(retrieved 15 Dec. 2025)

work page 2025

[5] [5]

S. H. Marshall and A. R. Perlis, Heronian tetrahedra are lattice tetrahedra, Amer. Math. Monthly,120(2), 2013, 140–149

work page 2013

[6] [6]

D. M. Y. Sommerville,An Introduction to the Geometry ofnDimensions, Dover, New York, 1958

work page 1958

[7] [7]

Stein, A note on the volume of a simplex, Amer

P. Stein, A note on the volume of a simplex, Amer. Math. Monthly,73(3), 1966, 299–301. 12

work page 1966