Geometric proof of $QM \ge AM$ Prove by geometric reasoning that:
$$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$
The proof should be different than one well known from Wikipedia:

DISCLAIMER: I think I devised such proof (that is not published anywhere to my knowledge), but I believe it is too complex (and not that beautiful). It involves some ideas from another question. I may post it in next few days as an answer, but I am hoping someone will offer better, more beautiful, proof by then.


$\qquad\qquad 2 \left(\;\color{blue}{a}^2 + \color{red}{b}^2\;\right) \;\geq\; \left(\;\color{blue}{a}+\color{red}{b}\;\right)^2$
