# Pigeon-Hole Principle and 2d grid

Q:Consider the 2D grid with integer coordinates.Prove that if we take 5ve points on the grid then there exist two of the points whose average is also a point on the grid.

I understand the basic idea behind Pigeon Hole principle, but I just cannot see how to apply it to this problem.

Here is my attempt to solve it.

There is always two integers in the set of five whose sum is even since if their sum is odd then there is another integer number whose sum with one of these numbers is going to be even. I apply this logic to x and y coordinates.

I had another idea of defining my pigeons as the number of all possible averages of two points from 5 points, and my pigeon boxes as the average is an integer and the average is not an integer.

Can someone give me hint on the direction of solving this problem. Thank you.

Look at the parity of coordinates, there are only $4$ options: $$(\text{even},\text{even}), (\text{even},\text{odd}), (\text{odd},\text{even}), (\text{odd},\text{odd})$$ So at least $2$ of the points fall in the same category, meaning the differences of $x$ and $y$ coordinates are both even, so there's a midpoint.