The sad life of lattice triangles

TL;DR

This paper investigates lattice triangles with integer-coordinate vertices, revealing that their key centers (circumcenter, centroid, orthocenter) generally do not all lie on the lattice, except in trivial cases, and discusses related properties.

Contribution

It provides new insights into the positions of classical centers of lattice triangles, highlighting limitations on their lattice membership beyond trivial cases.

Findings

Circumcenter, centroid, and orthocenter rarely all lie on the lattice.

Trivial lattice triangles are exceptions where all centers are on the lattice.

Additional properties of these centers are explored.

Abstract

This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never all lie on the integer lattice. Several further observations are made concerning the circumcenter, centroid and orthocenter.