Dear MSE-community!
I have begun on a journey through Gamelin's Complex Analysis. In the first chapter is an exercise described in the Math.SE question "Show that the set $z$ satisfying $|z−z_0|=\rho|z−z_1|\quad(\rho\neq 1)$ is a circle".
The accepted answer is making a completion of squares:
$$\begin{align} |z|^2 - \frac{1}{1-\rho^2}\bar{z}(z_0 - \rho^2 z_1) + \frac{1}{1-\rho^2}z(\overline{z_0 - \rho^2 z_1}) &= \frac{\rho^2 |{z_1}|^2 - |{z_0}|^2}{1 - {\rho}^2} \tag{1}\\ \iff \qquad \left|z-\frac1{1-\rho^2}\left(z_0-\rho^2z_1 \right)\right|^2 &=\frac{\rho^2|z_1|^2-|z_0|^2}{1-\rho^2}+\left|\frac{1}{1-\rho^2}(z_0-\rho^2z_1)\right|^2 \tag{2} \end{align}$$
I have myself derived equality $(1)$, but I cannot understand how to fill in the details myself regarding the completion of squares.
I am writing this as a question because I have no one to ask about it, and I have too low reputation to comment on the original thread.
All help is appreciated, thanks!