arxiv: 2604.25956 · v1 · submitted 2026-04-27 · 🧮 math.GM

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Lattice triangles whose centers are lattice points

Christian Aebi , Grant Cairns

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:21 UTC · model grok-4.3

classification 🧮 math.GM

keywords lattice trianglesorthocenterlattice pointsacute triangleslattice perimetercircumcentercentroid

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The pith

Acute lattice triangles have a lattice-point orthocenter precisely when their lattice perimeter is 6 or at least 8.

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

The paper proves an exact if-and-only-if criterion: acute triangles with vertices at lattice points and all angles less than 90 degrees admit a lattice-point orthocenter exactly when the sum of their Euclidean side lengths equals 6 or is at least 8. Parallel characterizations are given for the circumcenter and the centroid. These conditions are contrasted with the existence patterns that hold for obtuse and right lattice triangles. A reader cares because the result completely classifies the discrete perimeters at which the classical centers of a grid triangle can themselves lie on the grid.

Core claim

For any integer ℓ there exists an acute integer lattice triangle of lattice perimeter ℓ whose orthocenter is a lattice point if and only if ℓ = 6 or ℓ ≥ 8. Analogous if-and-only-if statements hold when the orthocenter is replaced by the circumcenter or the centroid. The existence thresholds for obtuse and right lattice triangles are different.

What carries the argument

The lattice perimeter (sum of Euclidean side lengths of a triangle whose vertices are integer lattice points) together with the geometric condition that the orthocenter coincides with a lattice point.

If this is right

Acute lattice triangles with lattice orthocenter exist for every integer perimeter at least 8.
The same perimeter thresholds govern when the circumcenter and centroid are lattice points.
Obtuse and right lattice triangles obey different existence rules for lattice centers.
For every sufficiently large integer perimeter at least one acute lattice triangle realizes a lattice orthocenter.

Where Pith is reading between the lines

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

The classification supplies a practical way to enumerate all minimal-perimeter examples with each center type.
Similar perimeter thresholds may exist for other triangle centers such as the incenter.
The distinction between acute and non-acute cases suggests that angle constraints interact strongly with lattice-point conditions.

Load-bearing premise

The triangles are non-degenerate with positive area, vertices strictly at lattice points, acute means all angles strictly less than 90 degrees, and the lattice perimeter equals exactly the given integer using Euclidean lengths.

What would settle it

An acute lattice triangle with lattice perimeter exactly 7 whose orthocenter is a lattice point, or the non-existence of any acute lattice triangle with lattice orthocenter for some specific ℓ ≥ 8.

Figures

Figures reproduced from arXiv: 2604.25956 by Christian Aebi, Grant Cairns.

**Figure 1.** Figure 1: A lattice triangle for which the circumcenter F, centroid G, and orthocenter H are lattice points. and thus n divides ℓ0. Hence n divides gcd(ℓ0, ℓ1, ℓ2) and thus gcd(ℓ1, ℓ2) = gcd(ℓ0, ℓ1, ℓ2). By the same reasoning, gcd(ℓi , ℓj ) = gcd(ℓ0, ℓ1, ℓ2), for all i ̸= j. □ Let us now recall the formulas for the circumcenter F, the centroid G, and the orthocenter H. Consider the triangle T with vertices 0, A = (… view at source ↗

**Figure 2.** Figure 2: Triangle with (ℓ0, ℓ1, ℓ2) = (2, 3, 5) and (m0, m1, m2) = (1, 2, 1). Let the vertices of the triangle be denoted V0, V1, V2, such that ℓi is the lattice length of the side Vi+1Vi+2 opposite Vi , where the subscripts are computed modulo 3. Recall the meaning of the numbers m0, m1, m2 from Lemma 2: • 2m0 is the lattice length of the segment from H to V0. • m1 is the lattice length of the segment from H to V1… view at source ↗

**Figure 3.** Figure 3: Acute triangle model. has lattice length gcd(ℓ − 6,(ℓ − 2)3k ) = 1. Thus T = OAB has lattice perimeter ℓ, independent of the choice of k. Note that ℓ ≥ 7 and hence x < z. And x + z and y are both multiples of 3, as required. (ii) If ℓ ≡ 4 (mod 6), set z = ℓ/2, x = 1, y = 1 2 (ℓ − 2)3k , for sufficiently large k. Then OA has lattice length ℓ/2, OB has lattice length gcd(1, 1 2 (ℓ − 2)3k ) = 1, and AB has la… view at source ↗

**Figure 4.** Figure 4: The incenter of the 3, 4, 5 right triangle. • ℓ = 21 : the triangle O,(15, 0),(3, 9) has centroid (6, 3) and orthocenter (3, 4). Case (b). We claim that if ℓ ∈ N, then there exists an acute lattice triangle of lattice perimeter ℓ, for which F, G, H ∈ Z 2 , if and only if ℓ is a multiple of 6 and ℓ ≥ 12. Moreover, we claim that this statement remains true if the word “acute” is replaced by “obtuse”, or by “… view at source ↗

**Figure 5.** Figure 5: A lattice triangle with its incenter on Z 2 but irrational inradius. a = |A|, b = |B|, c = |A − B|, the incenter of T is given by the following formula. (5) I = aB + bA a + b + c . A cursory examination of examples indicates that for the incenter there should be results analogous to those of Theorems 1, 2, 3, but with different numbers. The authors of this paper have not undertaken a study of this problem,… view at source ↗

read the original abstract

We show that for an integer $\ell$, there exists an acute integer lattice triangle of lattice perimeter $\ell$ such that its orthocenter is an integer lattice point, if and only if $\ell=6 $ or $\ell\ge 8$. Analogous results are obtained for the circumcenter and the centroid, and the results are contrasted with those for obtuse and right triangles.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves that for any integer ℓ, there exists an acute lattice triangle (vertices at integer lattice points, all angles strictly acute) with lattice perimeter ℓ (sum of Euclidean side lengths equal to ℓ) whose orthocenter is also a lattice point if and only if ℓ=6 or ℓ≥8. Analogous if-and-only-if characterizations are given for the circumcenter and centroid. The results are contrasted with the corresponding (different) conditions for obtuse and right lattice triangles.

Significance. If the proofs hold, the manuscript delivers clean, complete existence characterizations for three standard triangle centers in the setting of acute lattice triangles with integer perimeter. This is a precise, falsifiable contribution to lattice geometry that resolves the possible perimeters for which such centered triangles exist, and the contrast with obtuse/right cases adds comparative value. The iff structure and restriction to acute triangles are strengths.

minor comments (3)

§1: The definition of 'lattice perimeter' as the sum of Euclidean lengths equaling an integer ℓ should be stated explicitly with a short example (e.g., the ℓ=6 case) to avoid any ambiguity with Manhattan or other lattice metrics.
§3 (orthocenter case): The proof of the 'only if' direction for ℓ=7 relies on exhaustive enumeration of small-perimeter acute triangles; a brief remark on why no further cases arise for larger odd ℓ< some bound would improve readability.
Figure 2: The plotted example triangle for the circumcenter case has vertices that are hard to read at the given scale; adding coordinate labels would help verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states an external existence theorem: for integer ℓ there exists an acute lattice triangle of lattice perimeter ℓ with orthocenter (or other centers) at a lattice point, iff ℓ=6 or ℓ≥8. No equations, parameters, or derivations are exhibited in the abstract or described context that reduce the claimed result to a fitted input, self-definition, or self-citation chain. The definitions of lattice vertices, Euclidean perimeter summing to integer ℓ, and strict acuteness are standard and independent of the existence claim. The central result is a clean iff statement whose verification would rely on case analysis or enumeration external to any internal fit, making the derivation self-contained against the given benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard Euclidean geometry axioms for triangles, the definition of lattice points as Z², and the existence of altitudes and centers; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)

standard math Euclidean plane geometry holds for distances and angles between lattice points
Invoked implicitly when defining acute angles, orthocenter, and perimeter.
standard math Lattice points are points with integer coordinates in the plane
Core definition used throughout the statement.

pith-pipeline@v0.9.0 · 5339 in / 1290 out tokens · 68919 ms · 2026-05-07T17:21:59.609993+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages

[1]

A census of convex lattice polygons with at most one interior lattice point

Rabinowitz S. A census of convex lattice polygons with at most one interior lattice point. Ars Combin. 1989;28:83–96

1989
[2]

Reflexive polytopes in dimension 2 and certain relations in SL 2(Z)

Hille L, Skarke H. Reflexive polytopes in dimension 2 and certain relations in SL 2(Z). J. Algebra Appl. 2002;1:159–173

2002
[3]

Lattice polygons with two interior lattice points, Translation of Mat

Wei X, Ding R. Lattice polygons with two interior lattice points, Translation of Mat. Zametki91 (2012), no. 6, 920–933. Math. Notes 2012;91:868–877

2012
[4]

Algebraic capacities

Wormleighton B. Algebraic capacities. Selecta Math. (N.S.) 2022;28:Paper No. 9, 45

2022
[5]

Lecture Notes on Lattice Polytopes.https://www2.mathematik

Haase C, Nill B, Paffenholz A. Lecture Notes on Lattice Polytopes.https://www2.mathematik. tu-darmstadt.de/~paffenholz/daten/preprints/20210628_Lattice_Polytopes.pdf. Technische Universit¨ at Berlin; 2021

work page arXiv 2021
[6]

The fascination of the elementary

Scott PR. The fascination of the elementary. Amer. Math. Monthly 1987; 94:759–768

1987
[7]

Lattice polygons and the number 2i+7

Haase C, Schicho J. Lattice polygons and the number 2i+7. Amer. Math. Monthly 2009;116:151–165

2009
[8]

Integer area dissections of lattice polygons via a non-abelian Sperner’s Lemma

Abrams A, Pommersheim J. Integer area dissections of lattice polygons via a non-abelian Sperner’s Lemma. Amer. Math. Monthly 2025;132:737–753

2025
[9]

Lattice polygons and the number 12

Poonen B, Rodriguez-Villegas F. Lattice polygons and the number 12. Amer. Math. Monthly 2000; 107:238–250

2000
[10]

Icons of mathematics, an exploration of twenty key images

Alsina C, Nelsen RB. Icons of mathematics, an exploration of twenty key images. Dolciani Mathe- matical Expositions vol. 45. Washington DC: Mathematical Association of America; 2011

2011
[11]

Charming proofs, a journey into elegant mathematics

Alsina C, Nelsen RB. Charming proofs, a journey into elegant mathematics. Dolciani Mathematical Expositions vol. 42. Washington DC: Mathematical Association of America; 2010. Coll`ege Calvin, Geneva, Switzerland 1211 Email address:christian.aebi@edu.ge.ch Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia 3086 Emai...

2010