complex analysis – Mobius transformation that sends unit disk to unit disk


Suppose I have mobius transformation that sends to unit disk to unit disk. Then the mobius transformation would have a form of $f
_a(z)=frac{e^{i theta}(z-a)}{1- bar {a}z}$
where $a$ is a point in unit disk. But what if we put more constraint on this mobius transformation? What if we have four distinct points $x_1, x_2, x_3,$ and $x_4$inside the unit disk that sends $x_1mapsto x_2$ and $x_3mapsto x_4$? I know that there is an iff relation: $|f_{x_3}(x_1)|=|f_{x_4}(x_2)|$ iff $x_1, x_2, x_3,$ and $x_4$inside the unit disk that sends $x_1mapsto x_2$ and $x_3mapsto x_4$. But how can I see this?

I searched through popular complex analysis books like Ahlfors but there weren’t many facts about mobius transformation. In addition to the idea of this statement, it would be really nice if you could suggest me for further reading to study mobius transformation. Thanks.