Let $T$ be a Mobius transformation where $T(z_1)=z_1$ and $T(z_2)=z_2$ for some $z_1,z_2\in\mathbb{C}$. Suppose $z_3$ is the center of the segment $[z_1,z_2]$. Describe the possible image of $z_3$ under $T$.
The solution is very trivial so I wanted to verifying it because too simple solution may indicate I missed something.
A mobius tranformation is uniquely determined by $3$ elements of $\mathbb{C}\cup\{\infty\}$ so $T(z_3)$ could be any element in $\mathbb{C}\cup\{\infty\}$.