# Pigeonhole Principle and Geometry

Consider any five points in the plane that have integer coordinates: -Prove that there are two points such that the midpoint of the line segment joining those two points also has integer coordinates

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HINT: Think parity. The midpoint of $\langle a,b\rangle$ and $\langle r,s\rangle$ is $\left\langle\frac{a+r}2,\frac{b+s}2\right\rangle$. What does it take for both $a+r$ and $b+s$ to be even?
Any point with integer coordinates can classified to one and only one of the followings: $(e,e),(e,o),(o,e)$ or $(o,o)$, where $e$ is even, and $o$ is odd. But since you have 5 points in the plane...