Question 63
Consider the distinct complex numbers $a, b, c, d$. Prove that the following are equivalent:
i) For any $z \in \mathbb{C}$ we have $|z-a| + |z-b| \ge |z-c| + |z-d|$.
ii) There is $t \in (0, 1)$ such that $c = ta + (1-t)b$ and $d = (1-t)a + tb$.
i) For any $z \in \mathbb{C}$ we have $|z-a| + |z-b| \ge |z-c| + |z-d|$.
ii) There is $t \in (0, 1)$ such that $c = ta + (1-t)b$ and $d = (1-t)a + tb$.