A complex no z is said to be unimodular if $\left|z\right| = 1$ suppose ${\mathrm{z}}_1 \mathrm{and} {\mathrm{z}}_2$ are complex numbers such that $\frac{{\mathrm{z}}_1 - 2{\mathrm{z}}_2}{2- {\mathrm{z}}_1-{\mathrm{z}}_2}$ is unimodular and $z_2$ is not unimodular than $z_1$ has on a 

$$\begin{gathered} \left|\frac{z_1-2z_2}{2-z_1{\bar{z}}_2}\right|=1 \\ \Rightarrow {\left|z_1-2z_2\right|}^2={\left|2-z_1{\bar{z}}_2\right|}^2 \\ \Rightarrow \left(z_1-2z_2\right)\left({\bar{z}}_1-2{\bar{z}}_2\right)=\left(2-z_1{\bar{z}}_2\right)\left(2-z_1{z}_2\right) \\ \Rightarrow {\left|z_1\right|}^2-2z_1{\bar{z}}_2-2z_2{\bar{z}}_1+4{\left|z_2\right|}^2=4-2{\bar{z}}_1{z}_2-2z_1\bar{z}+\left|z_1\right.>{\left|z_2\right|}^2 \\ \Rightarrow {\left|z_1\right|}^2-4+{\left.4z_2\right|}^2-{\left|z_1\right|}^2{\left|z_2\right|}^2=0 \end{gathered}$$$$\begin{gathered} \Rightarrow \left({\left|{\mathrm{z}}_1\right|}^2-4\right)-{\left|{\mathrm{z}}_2\right|}^2\left({\left|{\mathrm{z}}_1\right|}^2-4\right)=0 \\ \Rightarrow \left(1-{\left|{\mathrm{z}}_2\right|}^2\right)\left({\left|{\mathrm{z}}_1\right|}^2-4\right)=0 \\ \Rightarrow \left|{\mathrm{z}}_2\right|\ne 1\Rightarrow {\left|{\mathrm{z}}_1\right|}^2=4 \\ \Rightarrow \left|{\mathrm{z}}_1\right|=2 \end{gathered}$$