sterographic projection of complex plane

Question

How author arrives at last step in following picture? How author arrives at $t=\frac{|z|^2}{1+|z|^2}$.

My attempt: since we see in pic that $(1-t)^2|z|^2=t(1-t)$ from this we conclude that, $t=(1-t)|z|^2$ but on other hand author arrives at $t=\frac{|z|^2}{1+|z|^2}$ How? Please help

Answer 1

From $(1−t)^2|z|^2=t(1−t)$ divide the whole equation by $(1-t)$ just like you did! :)

$(1−t)^2|z|^2=t(1−t)$ /:$(1-t)$

$|z|^2 (1-t)= t$

Now multiply the brackets on the left side and you get:

$|z|^2 - t|z|^2 = t$

$|z|^2 = t + t|z|^2$

$ |z|^2 = t(1+|z|^2)$

And finally:

$ t = |z|^2 / (1+|z|^2)$