9. Planar vectors $\boldsymbol{a}, \boldsymbol{b}$ have an angle of $\frac{\pi}{3}$ between them. If $|\boldsymbol{a}|,|\boldsymbol{b}|,|\boldsymbol{a}+\boldsymbol{b}|$ form an arithmetic sequence, find $|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{a}+\boldsymbol{b}|$.
Let $|\boldsymbol{a}|=x,|\boldsymbol{b}|=y,|\boldsymbol{a}+\boldsymbol{b}|=z$, then $z^{2}=x^{2}+y^{2}+2 x y \cos \frac{\pi}{3}=x^{2}+y^{2}+x y$ $\Rightarrow(2 y-x)^{2}=4 y^{2}+x^{2}-4 x y=x^{2}+y^{2}+x y \Rightarrow 3 y^{2}=5 x y \Rightarrow y=\frac{5}{3} x$
$\Rightarrow z^{2}=x^{2}+\frac{25}{9} x^{2}+\frac{5}{3} x^{2}=\frac{49}{9} x^{2} \Rightarrow z=\frac{7}{3} x$. Therefore, $x: y: z=3: 5: 7$.