Solution
= 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 satisfies
$$
|z-a|+|z-b|=|a-b|.
$$
The solution set is therefore precisely that segment in the <complex plane>.