The system of equations:

Question

|z-2-2i|=sqrt(23),
|z-8-5i|=sqrt(38)

has two solutions z_1 and z_2 in complex numbers. Find (z_1+z_2)/2.

My first instinct on this problem was to square each equation given that the square of the magnitude of a complex number is much easier to deal with than its magnitude alone.

I hoped to be able to then substitute in for z, but I quickly ran into problems given that the magnitude squared is the complex number times the conjugate of that number.

I don't know how to express these magnitudes squares' in terms of z and the conjugate. Any help is much appreciated :)