## 3. Solve the system of equations

$$
x: y: z=(y-z):(z-x):(x-y)
$$

where $x, y, z$ are distinct complex numbers.
Solution. Let

$$
\frac{x}{y-z}=\frac{y}{z-x}=\frac{z}{x-y}=k
$$

Then it follows that $k \neq 0, x y z \neq 0$ and $x=k(y-z), y=k(z-x), z=k(x-y)$. From the equations $y=k(z-x), z=k(x-y)$, we get

$$
y=\frac{k^{2}-k}{k^{2}+1} x, \quad z=\frac{k^{2}+k}{k^{2}+1} x
$$

and by substituting into $x=k(y-z)$, we obtain $\left(3 k^{2}+1\right) x=0$. Since $x \neq 0$, from the last equation it follows that $3 k^{2}+1=0$, i.e., $k= \pm \frac{i}{\sqrt{3}}$. For $k=-\frac{i}{\sqrt{3}}$, we get $y=\frac{-1+i \sqrt{3}}{2} x$, $z=-\frac{1+i \sqrt{3}}{2} x$, and for $k=\frac{i}{\sqrt{3}}$, we get $y=-\frac{1+i \sqrt{3}}{2} x, z=\frac{-1+i \sqrt{3}}{2} x$. It is easy to verify that the triples

$$
\left(x, \frac{-1+i \sqrt{3}}{2} x,-\frac{1+i \sqrt{3}}{2} x\right) \text { and }\left(x,-\frac{1+i \sqrt{3}}{2} x, \frac{-1+i \sqrt{3}}{2} x\right)
$$

where $x$ is any non-zero complex number, are solutions to the given system.