Try to place five points in (or on the perimeter of) an equilateral triangle with side length 10, as to maximize the distance between the points.
What is the greatest possible distance between the two closest points?
Solution
If you divide the equilateral triangle by connecting the midpoints of each side, you get four smaller, identical equilateral triangles with side length 5.
Since there are four small triangles and five points to be placed, two of the points must be in (or on) the same small triangle – a simple application of the Pigeonhole Principle.
The greatest possible distance between two points in any of the small triangles is the side length (5), and so the greatest possible distance between the closest two of the five points in the larger triangle can be at most 5 – and in fact this maximum of 5 is achievable (for example, placing the five points on five distinct vertices of the small triangles).