I'm trying to understand the following claim: If $z_1,z_2,z_3,z_4$ are points (as complex numbers) on a circle, then $z_1,z_3,z_4$ and $z_2,z_3,z_4$ determine the same orientation iff $CR(z_1,z_2,z_3,z_4)>0$.
Why is this?
This was the explanation I tried to explain to myself, but I don't know if it's fully correct, since I make a lot of assumptions to simplify the work.
Since the cross ratio is invariant under transformation, we can assume that $z_1,z_2,z_3,z_4$ lie on the real axis. Moreover, we can use a transformation to assume that $z_2=0$, $z_3=1$, and $z_4=2$. Now $ (z_1,z_2,z_3,z_4)=\frac{2(z_1-z_3)}{z_1-z_4} $ and so $(z_1,z_2,z_3,z_4)>0$ if any only if $\frac{z_1-z_3}{z_1-z_4}>0$. Now note that for any $z$, $ (z,z_1,z_3,z_4)=\frac{(z_1-z_4)z-(z_1-z_4)}{(z_1-z_3)z-2(z_1-z_3)} $ and $ (z,z_2,z_3,z_4)=\frac{2z-2}{z-2}. $ So the determinant of the first transformation is $-(z_1-z_3)(z_1-z_4)$, and that of the latter is $-2$. But $\frac{z_1-1}{z_1-2}>0$ when numerator and denominator have the same sign, that is, either when $z_1>z_3$ and $z_1>z_4$, or when $z_1