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Consider the inversion with respect to the sphere $S^n \subset R^{n+1}$, that is the map $$ \rho \colon x = (x_1,\dots x_{n+1}) \in R^{n+1}-\mathrm{O} \mapsto \frac{x}{\left \| x \right \|^2} \in R^{n+1}-\mathrm{O}.$$ If $x$, $y$ and $z$ are three nonzero points in $ R^{n+1}$, then the angle between the segments $yx$ and $yz$ is equal (in magnitude) to the angle formed by $\rho(y)\rho(x)$ and $\rho(y)\rho(z)$. I am looking for the simplest proof of this fact (basically that $\rho$ is anticonformal), possibly a really elementary one in which we don't make use of the transformation's Jacobian. Do you know any?

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    Why do you emphasize "in higher dimensions"? Everything happens in the space spanned by $x$, $y$ and $z$, so this reduces to the three-dimensional case.2011-11-14
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    I removed the (soft-question) tag. I don't think asking for the simplest proof of something counts as "not admitting a definitive answer".2011-11-14

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