Find all triples $(x,y, z)$ of real numbers such that $x^2 + y^2 + z^2 + 1 = xy + yz + zx + |x - 2y + z|$.
Source: 2010 Saudi Arabia BMO TST 1.1 - Balkan Math Olympiad
Tags: algebra
Find all triples $(x,y, z)$ of real numbers such that $x^2 + y^2 + z^2 + 1 = xy + yz + zx + |x - 2y + z|$.