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Show that the equation $\left| \frac{z+1}{z-1} \right|=2$ represents a Circle of radius $ \frac{4}{3} $ with center at $( \frac{5}{3} ,0)$

Well I know that what I should get should be of the form $\left| z- z_{0} \right|= \frac {4}{3}$ where $z_{0} = \frac{5}{3} $

When I solve it though I get $\left| z + 1 \right|=1 $ ( I multiplied the top and the bottom by (z+1) then canceled and collected terms, my textbook doesn't involve an example with something in the denominator this way).

Any help on how to approach this would be much appreciated.

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    See answer at [Finding an alternate definition for a set of complex numbers](http://math.stackexchange.com/a/2091243/291201). – 2017-01-20

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HINT Squaring the equation and using the identity $|z|^2=z\overline{z}$ yields the equation $$ (z+1)(\overline{z}+1)=4(z-1)(\overline{z}-1). $$ Now try to get this in the form satisfied by $\left|z-\frac{5}{3}\right|=\frac{4}{3}$, or equivalently $\left(z-\frac{5}{3}\right)\left(\overline{z}-\frac{5}{3}\right)=\frac{16}{9}$.