Let $w \in \mathbb{C}$ be a number that is neither real nor purely imaginary. (a) Show that the equation $$\left|\frac{z-1}{z+1}\right| = \left|\frac{w-1}{w+1}\right|$$ defines a circle in the complex plane, which passes through $w$. Verify that the interval $]-1,1[$ intersects this circle at a unique point; we denote this point by $y$. We will express $y$ in terms of the number $$\lambda = \left|\frac{w-1}{w+1}\right|.$$ (b) Show the inequality $$\left|\frac{1-w}{1-y}\right| > 1.$$ (c) Show that the equation $$\left|\frac{z-w}{z-y}\right| = \left|\frac{1-w}{1-y}\right|$$ defines a circle in the complex plane, which passes through $1$ and through $-1$. Deduce that, for all $x \in [-1,1] \setminus \{y\}$, we have $$\left|\frac{w-x}{y-x}\right| \geqslant \left|\frac{w-1}{y-1}\right| = \left|\frac{w+1}{y+1}\right|$$
Let $w \in \mathbb{C}$ be a number that is neither real nor purely imaginary.
(a) Show that the equation
$$\left|\frac{z-1}{z+1}\right| = \left|\frac{w-1}{w+1}\right|$$
defines a circle in the complex plane, which passes through $w$. Verify that the interval $]-1,1[$ intersects this circle at a unique point; we denote this point by $y$. We will express $y$ in terms of the number
$$\lambda = \left|\frac{w-1}{w+1}\right|.$$
(b) Show the inequality
$$\left|\frac{1-w}{1-y}\right| > 1.$$
(c) Show that the equation
$$\left|\frac{z-w}{z-y}\right| = \left|\frac{1-w}{1-y}\right|$$
defines a circle in the complex plane, which passes through $1$ and through $-1$.
Deduce that, for all $x \in [-1,1] \setminus \{y\}$, we have
$$\left|\frac{w-x}{y-x}\right| \geqslant \left|\frac{w-1}{y-1}\right| = \left|\frac{w+1}{y+1}\right|$$