As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$
I did some research and I found that there is a proof (not exactly the this inequality but an useful identity) of this on geometry revisited book section 1.7. the identity is $$OH^2=9R^2-(a^2+b^2+c^2)$$
where $H$ is orthocenter and $O$ is circumcenter. the proof of this identity uses Stewart's theorem, Euler line and ... . I find the proof not very nice and a little bit brute force. I want to know is there any different proof for it? and what is the name of this inequality?