MathNet 2017-part-02 — 176 problems (EN–VI) Question 14

Let $z_1$ and $z_2$ be complex numbers such that $|z_1| = |z_2| = 1$, and let $a$ and $b$ be real numbers such that $a + b = 1$. Prove
$$
|a z_1 + b z_2| \ge \frac{1}{2} |z_1 + z_2|.
$$