I am reading a paper on arXiv taking the idea of Ford circles and putting it into three dimensions, but am having difficulty with the following lemma, at a very basic level. Most of the details aren't relevant to my issue. What's important is that $\omega$ is the root of unity $\frac{-1+\sqrt{3}i}{2}$.
Now, everything here works for me except for $\frac{\alpha}{\beta}=\frac{b+c+c\omega}{a+b+c}$. ${|\beta|}^2$ is in fact $a+b+c$, so $\frac{\alpha}{\beta}=\frac{\alpha\bar{\beta}}{{|\beta|}^2}=\frac{(x+y\omega)(u-v\omega)}{a+b+c}=\frac{xu+yu\omega-xv\omega+yv(1+\omega)}{a+b+c}=\frac{xu+yv+(yu-xv+yv)\omega}{a+b+c}$, since $-\omega^2=1+\omega$. Now, $b+c=xu+yv-xv$ and $c=yu-xv$, which are not the real and $\omega$ parts of $\alpha\bar{\beta}$. So this can't hold in the generality required. So I was wondering if there's something obvious I'm misunderstanding, or some obvious modification that can be made to $a,b,c,d$ to get it to work or if I should just give up on it.
The mistake is that you have $\beta=u+v\omega$ and that you continue with $\overline{\beta}=u-v\omega$, which is not correct. Note that $\overline{\omega}=\omega^2=-1-\omega$.