Every trangle including a unit square in an orthogonal lattice has the lattice point covering property.
Let ABC be a triangle and an interior point M. We denote by AM =x1, BM = x2, CM = x3. The distances of the point M from BC, CA, AB are denoted by p1, p2, p3. Erdos-Mordell inequality asserts:
x1 + x2 + x3 no less than 2[p1 + p2 + p3]
The above problem proposed by P.Erdos in the American Mathematical Monthly in 1935 and solved byI.J.Mordell and D.F.Borrow in 1937. Later many Mathematicians obtained solutions and for a long time the problem was in the air.
For some more solutions you can download my paper on the Erdos-Mordell Inequality.