Following in Yiu's Footsteps but on the Eisenstein Lattice

TL;DR

This paper extends Yiu's result by proving that triangles with vertices on the Eisenstein lattice can also be realized, similar to Heron triangles on the integer lattice.

Contribution

It establishes an analogous theorem for the Eisenstein lattice, expanding the understanding of lattice realizability of triangles.

Findings

Triangles with vertices on the Eisenstein lattice are realizable.

The result parallels Yiu's theorem for the integer lattice.

Provides a new geometric insight into Eisenstein lattice configurations.

Abstract

Paul Yiu proved that all Heron triangles are realizable on the integer lattice. We give an analogous result for triangles with vertices on the Eisenstein lattice.