$1)$ Đặt $a+ b+ c= w$. Ta có:
$\left ( a- wx \right )^{2}+ \left ( b- wy \right )^{2}+ \left ( c- wz \right )^{2}\geqq 0$ hay:
$\left ( a- wx \right )^{2}+ \left ( b- wy \right )^{2}+ \left ( c- wz \right )^{2}\geqq\left ( a- wx+ b- wy+ c- wz \right )^{2}$ hay:
$\left ( xy+ yz+ zx \right )w^{2}- \left ( ay+ bz+ cx+ za+ xb+ yc\right )w+ ab+ bc+ ca\geqq 0$ hay:
$\Delta _{w}= 4\left ( ab+ bc+ ca \right )\left ( xy+ yz+ zx \right )- \left ( ay+ bz+ cx+ za+ xb+ yc\right )^{2}\leqq 0$ hay:
$ax+by+cz+2\sqrt{\prod\limits_{cyc}ab\prod\limits_{cyc}xy} \leqq x+y+z$ (quod erat demonstrandum)