Let $A$ and $B$ be on the unit circle.

Find a simple complex equation of the line $(A B)$.
$$
\begin{aligned}
Z \in(B C) & \Leftrightarrow \frac{z-b}{a-b} \in \mathbb{R} \Leftrightarrow \frac{z-b}{a-b}=\frac{\bar{z}-\bar{b}}{\bar{a}-\bar{b}} \Leftrightarrow(z-b)(\bar{a}-\bar{b})=(a-b)(\bar{z}-\bar{b}) \\
& \Leftrightarrow(z-b)\left(\frac{1}{a}-\frac{1}{b}\right)=(a-b)\left(\bar{z}-\frac{1}{b}\right) \Leftrightarrow(z-b)(b-a)=a b(a-b)\left(\bar{z}-\frac{1}{b}\right) \\
& \Leftrightarrow z=a+b-a b \bar{z}
\end{aligned}
$$