How to prove that the max distance between 5 points on a unit square perimeter is 0.75

Question

5 Ants are walking across a squared window, with the side length of 1 meter. I need to prove that in any given moment there are at least two ants the the distance between them is less than 75 cm.
I simplified the problem to the title: "How to prove that the max distance between 5 points on a unit square perimeter is 0.75", But I need to prove it without using calculus, I need to use Pigeonhole principle.

Answer 1

Divide the unit square into $4$ small squares of side $0.5$.

According to the Pigeonhole principle, there are at least 2 points in the same small square.

The maximum distance between two points in the small square is the distance of the diagonal, that is equal to $\sqrt{2}\cdot \frac{1}{2} \approx 0.71$ (cm) $< 0.75$ (cm)