Question 128
a) Complex numbers $x$ and $y$ are given on the perimeter of the unit circle such that
$$
\frac{\pi}{3} \le \arg(x) - \arg(y) \le \frac{5\pi}{3}.
$$
For each $z \in \mathbb{C}$ show that
$$
|z| + |z - x| + |z - y| \geq |zx - y|.
$$
b) Complex numbers $x$ and $y$ are given such that
$$
\frac{\pi}{3} \leq \arg(x) - \arg(y) \leq \frac{2\pi}{3}.
$$
For each $z \in \mathbb{C}$ show that
$$
|z| + |z - x| + |z - y| \geq \left| \frac{\sqrt{3}}{2}x + \left(y - \frac{x}{2}\right)i \right|.
$$
(For a complex number $z$, $\arg(z)$ is defined to be the counterclockwise angle between the axis of the real numbers with the vector $\vec{z}$.)
$$
\frac{\pi}{3} \le \arg(x) - \arg(y) \le \frac{5\pi}{3}.
$$
For each $z \in \mathbb{C}$ show that
$$
|z| + |z - x| + |z - y| \geq |zx - y|.
$$
b) Complex numbers $x$ and $y$ are given such that
$$
\frac{\pi}{3} \leq \arg(x) - \arg(y) \leq \frac{2\pi}{3}.
$$
For each $z \in \mathbb{C}$ show that
$$
|z| + |z - x| + |z - y| \geq \left| \frac{\sqrt{3}}{2}x + \left(y - \frac{x}{2}\right)i \right|.
$$
(For a complex number $z$, $\arg(z)$ is defined to be the counterclockwise angle between the axis of the real numbers with the vector $\vec{z}$.)