## Task B-4.2.

In the complex plane, there is an equilateral triangle $ABC$. The complex number $z=\sqrt{3}+4i$ is associated with vertex $C$, and the complex number $z=i$ is associated with the center $S$ of the circumscribed circle of the triangle. Determine the complex numbers associated with vertices $A$ and $B$.
## Solution.

The coordinates of point $C$ are $(\sqrt{3}, 4)$, and point $S(0,1)$. The radius of the circumcircle of the triangle is

$$
R=\sqrt{\left((0-\sqrt{3})^{2}+(1-4)^{2}\right)}=\sqrt{12}=2 \sqrt{3}
$$

Then, the length of the side of the given triangle is calculated from $R=\frac{a \sqrt{3}}{3}$, and it is $a=\frac{3 R}{\sqrt{3}}=6$.

Since vertices $A$ and $B$ lie on the circumcircle of triangle $ABC$, their coordinates satisfy the equation $x^{2}+(y-1)^{2}=12$.

Also, from $|AC|=6$ we get $(x-\sqrt{3})^{2}+(y-4)^{2}=36$.

By subtracting these two equations and simplifying, we obtain $y=-1-\frac{\sqrt{3}}{3} x$. Substituting this into the first equation, we get the quadratic equation $x^{2}+\sqrt{3} x-6=0$ whose solutions are $x_{1}=-2 \sqrt{3}$ and $x_{2}=\sqrt{3}$. Then, $y_{1}=1$ and $y_{2}=-2$. Therefore, vertex $A$ is associated with the complex number $2 \sqrt{3}+i$, and vertex $B$ with the complex number $\sqrt{3}-2 i$.