Midpoints joining integers on a plane lattice

Question

How can you prove that if five nodes of a plane lattice are chosen at random then, the midpoint of the segment between the two points is a lattice point.

Answer 1

Suppose the lattice is generated by the two vectors $u$ and $v$.

The lattice consists of all points $au+bv$, where $a$ and $b$ are integers. Call the pair $(a,b)$ the coordinates of the lattice point $au+bv$.

Consider the first coordinates of the $5$ points. At least $3$ of them have the same parity (all $3$ are even or all $3$ are odd).

Now take $3$ points whose first coordinates have the same parity. Then at least $2$ of their second coordinates have the same parity.

Thus there are $2$ lattice points $au+bv$ and $cu+dv$ such that $a$ and $c$ have the same parity, and $b$ and $d$ have the same parity. It follows that $\frac{a+c}{2}$ and $\frac{b+d}{2}$ are integers, and therefore $\frac{1}{2}\left((au+bv)+(cu+dv)\right)$ is a lattice point.