On lattice triangles satisfying $\boldsymbol{B(T)=3}$ with collinear interior lattice points

TL;DR

This paper studies specific lattice triangles with three boundary points and a certain number of interior points, focusing on when all interior boundary points are collinear, and characterizes the integers for which this property holds.

Contribution

It characterizes the integers k for which all lattice triangles with 3 boundary points and k interior points have collinear boundary points.

Findings

Identifies the set of integers k with the collinearity property.

Provides conditions under which the property holds for lattice triangles.

Contributes to the understanding of lattice triangle configurations.

Abstract

A lattice point in $\mathbb{R}^2$ is a point $(x,y)$ with $x,y\in\mathbb{Z}$, and a lattice triangle is a triangle whose three vertices are all lattice points. We investigate the integers $k$ with the property that if $T$ is a lattice triangle with $3$ boundary points and $k$ points in the interior, then all $k$ boundary points must be collinear.