Question 121
Let $x, y, z$ be complex numbers so that $x \neq z$ and $\frac{x-y}{x-z} \notin \mathbb{R}$. Prove that
$$ \min(|2x - y - z|, |2y - x - z|, |2z - x - y|) \le \frac{1}{\sqrt{3}}(|x - y| + |y - z| + |z - x|). $$
$$ \min(|2x - y - z|, |2y - x - z|, |2z - x - y|) \le \frac{1}{\sqrt{3}}(|x - y| + |y - z| + |z - x|). $$