White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:

If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $T$, then the two following conditions are equivalent:

(1) The only points of $\Lambda$ in $T$ other than the vertices lie on a pair of opposite edges of $T$;

(2) There is a pair of parallel lattice planes of $\Lambda$ through a pair of opposite edges of $T$ such that no points of $\Lambda$ lie between these planes.

His proof is rather complicated. Has anybody proved this theorem in a simpler way? What proof is the simplest one?