Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in \mathbb{R}^k$ and $r > 0$ such that, for all $\mathbf{x} \in \mathbb{R}^k$, we have $$\vert \mathbf{x} - \mathbf{a} \vert = 2 \vert \mathbf{x} - \mathbf{b} \vert$$ if and only if $$\vert \mathbf{x} - \mathbf{c} \vert = r.$$
Although Rudin has given a solution, namely $3\mathbf{c} = 4 \mathbf{b} - \mathbf{a}$, $3r = 2\vert \mathbf{b} - \mathbf{a} \vert$, I'm wondering how he's obtained it.
How to attack this type of a problem?
Is this problem part of the exercises by some well-thought-out design? I mean is it going to be used later on in the book? Or, is it just to give the reader some practice?