This is the 2-dimensional case of Minkowski's theorem, and except of the lack of imagination for dimensions $>3$, I see no reason why this case is easier. The $n$-dimensional version is: Given a convex region $R$ symmetrical to the origin in the $n$-dimensional space. Show that if $R$ has volume greater than $2^n$, then $R$ contains a lattice point different from the origin.