Complex complication

$z_1$ and $z_2$ are two constant complex numbers such that $\displaystyle | z_1 - z_2 | = 5$ . A complex number, $z$ , satisfies the following equation.

$\displaystyle | z + \sqrt{ z^2 + z_1 z_2 - zz_1 - zz_2 } - \frac{z_1 + z_2}{2} | + | z - \sqrt{ z^2 + z_1 z_2 - zz_1 - zz_2 } - \frac{z_1 + z_2}{2} | = 10$

The locus of another complex number $z_3$ is given as :
For the equation $\displaystyle \tan ( \text{ arg} ( z - z_3 ) ) = \tan \phi$ , there exists exactly one value of $z$ for two different values of $\phi$ , $( \phi _1$ and $\phi _2 )$ such that $\displaystyle | \phi _1 - \phi _2 | = m\pi + \frac{\pi }{2} \quad ; m \in \mathbb{Z}$ .
( The value of the complex number $z$ is not same for the two values of $\phi$ but the number of solutions is unity for the two values. )

The value $\displaystyle |2z_3 - z_1 -z_2 |$ is found to be constant. Enter the answer as the square of the constant value.

Details and Assumptions:

• $\text{arg }(z)$ gives the principal argument of a complex number $z$ .