Seven points are given inside a regular hexagon whose sides have length $1$. Prove that there are two among these seven points such that the distance between them is at most $1$.
Now if I divide the hexagon into $6$ regions, since we have $7$ points, by the pigeon hole principle there is a region with at least two points in it. The distance between two points is at most $1$ because each region is an equilateral triangle with sides of length $1$.
Would this be sufficient or any other approaches that would work better.
Best Answer
This seems sufficient, but you can add something like this:
The distance between two points in an equilateral triangle with side length $1$ is at most $1$ because this triangle lies in a circle of radius $1/2$.