Lattice triangles whose centers are lattice points

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.