Lemma 2

There is a vertex x of A such that the distance from x to B is equal to h (A, B).

Proof :

This is easily proved :  a line going from a vertex b of a triangle abc to some point that belongs to the opposite side ß is always shorter than one of ab or cb, or both.

So if some point x of ß is the furthest point of A from b, then x has to be an endpoint of ß, and thus a vertex of A.


Notice however that lemma 2 applies only to a, the furthest point of A relative to b :  as illustrated below, the closest point b of B relative to a might not be a vertex of B.