1. Point $A(3,0)$ is a fixed point on the circle $x^{2}+y^{2}=9$, take another two points $B$ and $C$ on the circle such that $\angle B A C=\frac{\pi}{3}$, find the equation of the locus of the centroid of $\triangle A B C$.
1. Let $G(x, y)$ be the centroid of $\triangle A B C$. Taking another $B$ and $C$ on the circle such that $\angle B A C=\frac{\pi}{3}$, then $\angle B O C=\frac{2 \pi}{3}$. By plane geometry, the trajectory equation of the midpoint $D\left(x_{0}, y_{0}\right)$ of the chord $B C$ is
$$
x_{0}^{2}+y_{0}^{2}=\left(\frac{3}{2}\right)^{2},-\frac{3}{2} \leqslant x_{0} \leqslant \frac{3}{4} .
$$

Given $\frac{A D}{D G}=-3$, we have $x_{0}=\frac{3 x-2}{2}, y_{0}=\frac{3 y}{2}$. Substituting $D$'s coordinates into the equation, we obtain $(x-1)^{2}+y^{2}=1$ (where $0 \leqslant x<\frac{3}{2}$ ).