In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote.
Lemma 1: Let the nodes of a triangle $K$ be $A$, $B$ and $C$, and the longest edge length of $K$ be $h$ (see attached figure). Then, the following inequality holds $$\min_{P \in K}\max(|PA|, |PB|, |PC|) \le \frac{\sqrt{3}}{3} h\:.$$ Moreover, the equality holds only for a regular triangle with $P$ being the circumcenter of $K$.
The 3D version of the above Lemma is as follows.
Lemma 2 For any tetrahedron $K$, the vertices of which are denoted by $A,B,C,D$, let $h$ be the longest edge length. Then, $$\min_{P \in K} \max (|PA|, |PB|,|PC|, |PD|) \le \frac{\sqrt{6}}{4} h $$ Moreover, the equality holds only for a regular tetrahedron $K$ with $P$ selected as the circumcenter of $K$;
This is well discussed in the Jung's theorem: https://en.wikipedia.org/wiki/Jung's_theorem