Theorem Given non-negative real numbers $a, b, c$ not all zero, for any real numbers $x, y, z$, the inequality
$$\lambda\left(x^{2}+y^{2}+z^{2}\right) \geqslant a x y+b y z+c z x$$

holds, where $\lambda=\sqrt{\frac{a^{2}+b^{2}+c^{2}}{3}} \cos \frac{\theta}{3}, \theta=\arccos \left[\frac{a b c}{\left(\left(a^{2}+b^{2}+c^{2}\right) / 3\right)^{32}}\right]$, equality holds if and only if
$$\frac{x}{a b+2 c \lambda}=\frac{y}{a c+2 b \lambda}=\frac{z}{4 \lambda^{2}-a^{2}} \text { . }$$