Only Law Of Cosines, Let $r^2=x^2 + y^2 - xy<1$. Its a triangle with angle $60$, and leghts $x,y,r$. After of this: Cases: $x=y$ or $x<y$

parmenides51 wrote: Let $x$ and $y$ be real numbers such that $x^2 + y^2 - 1 < xy$. Prove that $x + y - |x - y| < 2$. $x^2 + y^2 - 1 < xy$ $\iff$ $xy<1-(x-y)^2\le 1$ $\implies$ $\min(x,y)<1$ $\implies$ $x+y-|x-y|=2\min (x,y)<2$ Q.E.D.