Lattice triangles whose centers are lattice points

TL;DR

This paper characterizes the existence of acute integer lattice triangles with lattice orthocenters, circumcenters, and centroids based on their perimeter, revealing specific conditions for such triangles.

Contribution

It provides a complete characterization of when acute lattice triangles with lattice centers exist, based on their perimeter, and compares these results with obtuse and right triangles.

Findings

Acute lattice triangles with lattice orthocenters exist if and only if perimeter is 6 or at least 8.

Analogous conditions are established for circumcenters and centroids.

Results differ significantly for obtuse and right triangles.

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.