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According to remmert there is a relationship between the crossratios:

C(z,u,v,w) = \frac{(z-v)(u-w)}{(z-w)(u-v)} \text{ and } C(z,v,u,w)= \frac{(z-u)(v-w)}{(z-w)(v-u)} where $z,u,v,w \in \mathbb{C}$

I have tried multiplying things out to see a relationship, but I don't see anything. What could be meant with "relationship"?

Another question deals with the value of the crossratio: On which of the three arcs through $u,v,w$ must z lie so that $0 < C(z,u,v,w) < 1$?

How does one see that there must be three arcs through u,v,w without plotting and how can we characterize the cross ratio without plotting?

Thanks for every suggestion.

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    @anon I saw a tool once where you could put points in the plane and it would plot the circle through them and calculate the cross ratio also. Between u,v,w there must be three arcs, however I don't know how to solve the actual question with even knowing that. Il y a there were a lot of typed errors, I fix. Sorry for that.2012-02-13

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The cross ratio $C(\cdot,\cdot,\cdot,\cdot)$ is a function of four complex variables, no two of them equal. When the values of these variables are permuted the value $\lambda$ of $C$ undergoes a certain rational transformation, e.g., $\lambda\mapsto{\lambda\over\lambda-1}$, depending on the permutation chosen. The $24$ permutations produce $6$ (in special cases fewer) different values of $C$. For details see this Wikipedia article.