Consider 6 distinct points in a plane. Let $m$ and $M$ be the minimum and maximum distances between any pair of points. Show that $M/m \ge \sqrt3$.
I am more interested in arrangement of these points where the equality holds.
I figured out that the least possible value of $M/m$ should happen when the six points lie on corners and centre of a regular pentagon. But this gave me $2cos18 = \sqrt3.6$. How do I get $\sqrt3$?
Just a hint will suffice. Thanks in advance.
Community wiki answer so the question can be marked as answered: Apparently $2\cos\pi/10$ is indeed the minimal value of the ratio.