Proof using complex numbers Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$.
I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with $z = x + iy$ and $ = a+ib$ but I am getting stuck near finish.

Suppose $|z|=1$: then
$$
\left|\dfrac{z-w}{1-\bar{z}w}\right|=
\frac{1}{|z|}\left|\dfrac{z-w}{1-\bar{z}w}\right|=
\left|\dfrac{z-w}{z(1-\bar{z}w)}\right|=
\left|\dfrac{z-w}{z-z\bar{z}w}\right|=
\left|\dfrac{z-w}{z-w}\right|=1
$$
If $|w|=1$, consider that $|1-\bar{z}w|=|1-\bar{w}z|$ because they are conjugate.
