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Lattice triangles whose centers are lattice points

This paper establishes that an acute integer lattice triangle with a given integer lattice perimeter $\ell$ possesses an orthocenter, circumcenter, or centroid that is also a lattice point if and only if $\ell=6$ or $\ell\ge 8$ , while contrasting these findings with the properties of obtuse and right triangles.

Original authors: Christian Aebi, Grant Cairns

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Christian Aebi, Grant Cairns

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are drawing shapes on a giant sheet of graph paper where every intersection of the lines is a "dot." In the world of this paper, these dots are called lattice points. A lattice triangle is simply a triangle where all three corners land perfectly on these dots.

The authors of this paper are playing a game of "connect the dots" with a twist. They aren't just looking at the corners; they are looking at the triangle's "heart" or "center." In geometry, every triangle has three famous centers:

The Centroid (G): The balance point. If you cut the triangle out of cardboard, this is where you could balance it on your finger.
The Circumcenter (F): The center of a circle that passes through all three corners.
The Orthocenter (H): A point where lines drawn from the corners to the opposite sides (at right angles) all meet.

Usually, these centers land somewhere in the empty space between the dots. The big question the authors asked is: "Can we draw a triangle on the graph paper such that one of these special centers also lands exactly on a dot?"

But there's a catch. They only care about acute triangles (triangles where all angles are sharp, less than 90 degrees). They also measure the triangle's size not by inches or centimeters, but by counting how many dots are on the edges. They call this the "lattice perimeter."

The Big Discovery: The "Magic Numbers"

The authors found that for these special triangles to exist, the number of dots on the edge (the perimeter) has to follow very specific rules. It's like a lock that only opens with certain keys.

1. The Orthocenter (The "Right-Angle" Center)

The Rule: You can draw an acute triangle with a dot-center if the perimeter is 6 or 8 or higher.
The Missing Numbers: You cannot do this with perimeters of 3, 4, 5, or 7.
The Analogy: Imagine trying to build a tower with blocks. If you have 3, 4, 5, or 7 blocks, the tower is too wobbly to have a "perfect" center dot. But if you have 6 blocks, or 8 and up, the structure stabilizes, and the center lands perfectly on a dot.

2. The Circumcenter (The "Circle" Center)

The Rule: This one is pickier. The perimeter must be an even number. Furthermore, it must be 8 or 12 or higher.
The Missing Numbers: You can't do it with perimeters of 4, 6, or 10.
The Analogy: This center is like a tightrope walker. It needs a very specific even number of steps to stay balanced. If the perimeter is 6 or 10, the tightrope is too short or too long, and the walker falls off the dot.

3. The Centroid (The "Balance" Center)

The Rule: This is the most flexible one. You can do it with almost any number, as long as the perimeter is 3 or higher, except for 5 and 11.
The Missing Numbers: 5 and 11 are the "forbidden zones."
The Analogy: Think of the centroid as a seesaw. It's very easy to balance, but if you have exactly 5 or 11 people on it, the math just doesn't work out to land on a dot. Any other number works fine.

What About Other Triangles?

The authors also checked obtuse (one wide angle) and right (one 90-degree angle) triangles.

The Surprise: For these "non-sharp" triangles, the rules are much looser. You can almost always find a triangle with a dot-center, regardless of the perimeter size (with very minor restrictions for the circumcenter).
The Takeaway: The strict "magic numbers" (like 5, 7, 11) only apply to the sharp, acute triangles. The "lazy" triangles (obtuse and right) are much easier to fit onto the grid.

The "Double Trouble" Scenario

What if you want two or even all three of these centers to land on dots at the same time?

The authors found that if you want the Centroid and Orthocenter to both be dots, the perimeter must be a multiple of 3 (like 9, 12, 15...), but not 3 or 6.
If you want all three (Centroid, Circumcenter, and Orthocenter) to be dots, the perimeter must be a multiple of 6 (like 12, 18, 24...), but not 6.

The Open Mystery: The "Incenter"

Finally, the paper ends with a challenge for future mathematicians. They looked at the Incenter (the center of the circle that fits inside the triangle).

They suspect there are similar "magic numbers" for this center, but they haven't solved it yet.
They even found a weird example where the center is a dot, but the circle inside the triangle has a "weird" size (an irrational number), and the points where the circle touches the triangle aren't even dots.
The Call to Action: They are inviting others to take up the challenge of solving this specific puzzle, warning that it might be even harder than the problems they just solved.

Summary

In short, this paper is a mathematical treasure hunt. It maps out exactly which sizes of "sharp" triangles can be drawn on a grid so that their most important centers land perfectly on the grid intersections. It turns out that for sharp triangles, the universe has a very specific list of allowed sizes, excluding a few curious numbers like 5, 7, and 11.

1. Problem Statement

The paper investigates the intersection of lattice geometry (integer geometry) and Euclidean geometry. Specifically, it addresses the following question:
Given a specific type of triangle (acute, obtuse, or right) with vertices on the integer lattice $\mathbb{Z}^2$ , what are the possible values for its lattice perimeter $\ell$ if one of its classical centers (Circumcenter $F$ , Centroid $G$ , or Orthocenter $H$ ) is also a lattice point?

Definitions:

Lattice Point: An element of $\mathbb{Z}^2$ .
Lattice Length: For a segment connecting two lattice points, the lattice length is the number of lattice points on the segment minus one.
Lattice Perimeter ( $\ell$ ): The sum of the lattice lengths of the sides of the polygon. This is equivalent to the total number of lattice points on the boundary of the polygon.
Centers:
- $F$ : Circumcenter (intersection of perpendicular bisectors).
- $G$ : Centroid (intersection of medians).
- $H$ : Orthocenter (intersection of altitudes).

The authors focus primarily on acute triangles, as the conditions for obtuse and right triangles are shown to be less restrictive.

2. Methodology

The authors employ a combination of number-theoretic arguments, geometric constructions, and algebraic analysis of lattice invariants.

Algebraic Formulas: They utilize explicit coordinate formulas for $F$ , $G$ , and $H$ in terms of vertex coordinates. A key identity used is the Euler line relation: $2F + H = 3G$ .
Lattice Invariants:
- Lemma 1: Establishes that for any lattice triangle, the greatest common divisor (GCD) of any two side lengths equals the GCD of all three side lengths.
- Lemma 2 & 3: Relate the angles of the triangle to the lattice lengths of segments connecting vertices to the orthocenter. Specifically, if $H$ is a lattice point, $\tan(\theta) = \ell_{\theta}/m$ , where $\ell_{\theta}$ is the opposite side length and $m$ is the lattice length of the segment from the vertex to $H$ .
- Parity Arguments (Lemma 4 & Corollary 1): Analyze the parity (even/odd) of coordinates to determine constraints on the circumcenter. If $F$ is a lattice point, the vertices must share specific parity properties, forcing the lattice perimeter to be even.
- Modular Arithmetic (Lemma 7): If the centroid $G$ is a lattice point and one side length is a multiple of 3, then all side lengths must be multiples of 3.
Constructive Proofs: To prove existence for valid $\ell$ , the authors explicitly construct triangles with vertices $O(0,0)$ , $A(z,0)$ , and $B(x,y)$ satisfying the required conditions.
Exclusion Proofs: To prove non-existence for invalid $\ell$ , they enumerate all possible integer partitions of $\ell$ into three side lengths and demonstrate contradictions using the lemmas above (e.g., showing a triangle cannot be acute or that the GCD condition fails).
Numerical Verification: In the proof of Theorem 2, they use numerical calculations of arctangent sums to show that certain angle combinations cannot sum to $\pi$ (180 degrees), thereby ruling out specific lattice perimeters.

3. Key Contributions and Results

The paper establishes necessary and sufficient conditions for the existence of acute lattice triangles with a lattice center for each of the three classical centers.

A. The Orthocenter ( $H$ )

Theorem 1: An acute lattice triangle with lattice perimeter $\ell$ and a lattice orthocenter exists if and only if $\ell = 6$ or $\ell \geq 8$ .
Excluded Values: $\ell \in \{3, 4, 5, 7\}$ .
Insight: The exclusion of small integers like 7 is linked to a number-theoretic curiosity regarding the sum of three pairwise coprime integers.

B. The Circumcenter ( $F$ )

Theorem 2: An acute lattice triangle with lattice perimeter $\ell$ and a lattice circumcenter exists if and only if $\ell$ is even and ( $\ell = 8$ or $\ell \geq 12$ ).
Excluded Values: Even $\ell \in \{2, 4, 6, 10\}$ .
Constraint: The lattice perimeter must be even (a consequence of Corollary 1). The condition $\ell \neq 10$ is derived from complex angle analysis involving arctangents.

C. The Centroid ( $G$ )

Theorem 3: An acute lattice triangle with lattice perimeter $\ell$ and a lattice centroid exists if and only if $\ell \geq 3$ and $\ell \notin \{5, 11\}$ .
Excluded Values: $\ell \in \{1, 2, 5, 11\}$ .
Insight: The exclusions of 5 and 11 are related to a number-theoretic proposition involving sums of integers coprime to 3.

D. Combined Centers

The paper also analyzes cases where multiple centers are lattice points:

$G$ and $H$ are lattice points: $\ell$ must be a multiple of 3 and $\ell \geq 9$ (excluding 3 and 6).
$F, G, H$ are lattice points: $\ell$ must be a multiple of 6 and $\ell \geq 12$ (excluding 6).
Note: If $F$ is a lattice point, $H$ is automatically a lattice point (via the Euler line).

E. Obtuse and Right Triangles

The authors contrast these results with non-acute triangles:

Orthocenter: For obtuse/right triangles, any $\ell \geq 3$ is possible.
Circumcenter: For obtuse/right triangles, any even $\ell \geq 4$ is possible.
Centroid: The restriction $\ell \notin \{5, 11\}$ holds for obtuse triangles, but for right triangles, $\ell$ must be a multiple of 3 (and $\geq 9$ ).

4. Significance

Bridging Fields: The paper highlights the tension between the transformation groups of lattice geometry ( $GL(2, \mathbb{Z})$ ) and Euclidean geometry ( $O(2, \mathbb{R})$ ). It demonstrates how elementary geometric constraints can lead to non-trivial number-theoretic restrictions.
Curious Exclusions: The results reveal "gaps" in the set of possible lattice perimeters (e.g., why 7 is impossible for the orthocenter, or why 5 and 11 are impossible for the centroid). These gaps are not arbitrary but stem from deep properties of integer partitions and coprimality.
Open Problems: The paper concludes by posing an open problem regarding the Incenter. While the authors suspect similar results exist, the complexity of the incenter formula (involving side lengths in the denominator) suggests the problem may be significantly harder. They provide examples where the incenter is a lattice point but the inradius is irrational, illustrating the unique challenges of this case.

5. Conclusion

Aebi and Cairns provide a complete classification of lattice perimeters for acute lattice triangles with lattice centers. Their work transforms a geometric question into a solvable number-theoretic problem, identifying precise "forbidden" values for the perimeter and offering constructive proofs for all valid cases. The results are summarized in Table 3 of the paper, which serves as a comprehensive reference for achievable lattice perimeters under various center constraints.