Lemma 2There is a vertex

xof A such that the distance fromxto B is equal to h (A, B).

Proof :

This is easily proved:a line going from a vertexbof a triangleabcto some point that belongs to the opposite sideßis always shorter than one ofaborcb, or both.So if some point

xofßis the furthest point of A fromb, thenxhas to be an endpoint ofß, and thus a vertex of A.Q.E.D.

Notice however that lemma 2 applies only to

a, the furthest point of A relative tob: as illustrated below, the closest pointbof B relative toamight not be a vertex of B.