CZ-SK-MO 2007 Problem 6

Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$ . (It means, those three fractions make a permutation of $x, y$ , and $z$ .)