### Complex Numbers

goals
For two unimodular complex numbers $z_1$ and $z_2$, $\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1} is equal to For two unimodular complex numbers z_1 and z_2, \begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1}$ is equal to
$\dfrac { 5+2i }{ 1-i } +\dfrac { 5-2i }{ 1+i }$
Let $z_1$ = 18 + 83i, $z_2$ = 18 + 39i, ana $z_3$= 78 + 99i. where i = $\sqrt-1$. Let z be a unique comlpex number with the properties that $\dfrac{z_3 - z_1}{z_2 - z_1}$ $\cdot$ $\dfrac{z - z_2}{z - z_3}$ is a real number and the imaginary part of the size z is the greatest possible.
$\dfrac { 5+2i }{ 1-i } +\dfrac { 5-2i }{ 1+i }$