Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 1B / a / ii / Solution

Equality in the triangle inequality holds exactly when the two displacement vectors $a-z$ and $z-b$ point in the same direction. Thus $z$ lies on the closed line segment joining $a$ and $b$. Conversely, every point of that segment satisfiesThe solution set is therefore precisely that segment in the complex plane.