Example: In the complex plane, it is known that the vertices $A$, $B$ of the equilateral triangle $\triangle ABC$ represent the complex numbers 2 and $\frac{1}{2}+\frac{\sqrt{3}}{2} i$, respectively. Find the complex number that $C$ represents.
Let $C$ correspond to $z$, by the theorem
$$
\begin{array}{l}
\left\{\begin{array}{l}
2+\frac{1}{2}+\frac{\sqrt{3}}{2} i+z=3 z_{0}, \\
2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)+2 z+\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right) z=3 z_{0}^{2},
\end{array}\right. \\
\text { Solving, we get }\left\{\begin{array}{l}
z=\frac{1}{2}-\frac{\sqrt{3}}{2} i, \text { or }\left\{\begin{array}{l}
z=2+\sqrt{3} i, \\
z_{0}=1
\end{array}, \frac{3}{2}+\frac{\sqrt{3}}{2} i,\right.
\end{array}\right.
\end{array}
$$

The third vertex is $C\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)$ or $C(2+\sqrt{3} i)$.