8. Given that the argument of the complex number $z$ is $60^{\circ}$, and $|z-1|$ is the geometric mean of $|z|$ and $|z-2|$, find $|z|$.
8. $\left|z^{2}-2 z\right|=\left|z^{2}-2 z+1\right|$, so $\operatorname{Re}\left(z^{2}-2 z\right)=-\frac{1}{2}$ Let $z=\frac{1}{2} t+\frac{\sqrt{3}}{2} t \mathrm{i}$, so $\frac{1}{2} t^{2}+t-\frac{1}{2}=0, t=-1 \pm \sqrt{2}$, so $|z|=|t|=\sqrt{2} \pm 1$