Every point in a straight line are coloured using 2 colours. Show that there are 2 points and their midpoint (lying on this straight line) of the same colour.
I saw this question on a pigeon hole principle problem sheet and have found a solution that uses proof by contradiction.
Suppose there is no such pair of points of the same colour with a midpoint of the same colour. There must be at least 2 points on this line of the same colour. Define the distance between them as d. Also, their midpoint must be of the other colour otherwise there is a contradiction. The points that are d away on either side must also be of the other colour, otherwise there is a contradiction as well. However these two points of the other colour have a midpoint of the same colour, therefore there is a contradiction so there is always a pair of points with a midpoint of the same colour.
Can this be rewritten using pigeon hole principle? Also are there any changes I should make to the written solution to make it clearer?
Essentially you do use pigeon-hole "in the background":
Let $A,B,C$ be three distinct points. By pigeon-hole, two of them (say, $X,Y$) have the same colour. Let $Z$ be the midpoint of $X,Y$, let $U$ be the reflection of $X$ at $Y$ and $V$ the reflection of $Y$ at $X$. Then $(X,Z,Y)$, $(X,Y,U)$, $(V,X,Y)$, and $(V,Z,U)$ are all triples of "two points and their midpoint". Then $(V,Z,U)$ is monochromatic, or at least one of the three has the same colour as $X$ and $Y$ and makes one of the other three triples monochromatic