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Homeproof of converse of M\"obius transformation cross-ratio preservation theorem

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# proof of converse of Möbius transformation cross-ratio preservation theorem

Suppose that $a,b,c,d$ are distinct. Consider the transform $\mu$ defined as

$\mu(z)={(b-d)(c-d)\over(c-b)(z-d)}-{b-d\over c-b}.$ |

Simple calculation reveals that $\mu(b)=1$, $\mu(c)=0$, and $\mu(d)=\infty$. Furthermore, $\mu(a)$ equals the cross-ratio of $a,b,c,d$.

Suppose we have two tetrads with a common cross-ratio $\lambda$. Then, as above, we may construct a transform $\mu_{1}$ which maps the first tetrad to $(\lambda,1,0,\infty)$ and a transform $\mu_{2}$ which maps the first tetrad to $(\lambda,1,0,\infty)$. Then $\mu_{2}^{{-1}}\circ\mu_{1}$ maps the former tetrad to the latter and, by the group property, it is also a Möbius transformation.

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## Mathematics Subject Classification

30E20*no label found*

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