Here is Prob. 16, Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Suppose $k \geq 3$, $\mathbf{x}, \mathbf{y} \in \mathbb{R}^k$, $\vert \mathbf{x} - \mathbf{y} \vert = d > 0$, and $r > 0$. Prove:
(a) If $2r > d$, there are infinitely many $\mathbf{z} \in \mathbb{R}^k$ such that $$\vert \mathbf{z} - \mathbf{x} \vert = \vert \mathbf{z} - \mathbf{y} \vert = r.$$
(b) If $2r = d$, there is exactly one such $\mathbf{z}$.
(c) If $2r< d$, there is no such $\mathbf{z}$.
How must these statements be modified if $k$ is $2$ or $1$?
My work:
First of all, if an element $\mathbf{z} \in \mathbb{R}^k$ satisfies $$\vert \mathbf{z} - \mathbf{x} \vert = \vert \mathbf{z} - \mathbf{y} \vert = r,$$ then we have $$d = \vert \mathbf{x} - \mathbf{y} \vert \leq \vert \mathbf{z} - \mathbf{x} \vert + \vert \mathbf{z} - \mathbf{y} \vert = 2r.$$ So if $2r < d$, then there is no such $\mathbf{z} \in \mathbb{R}^k$, as Rudin asserts in part (c).
Now let's take a real number $\alpha \in (0,1)$, let $\mathbf{z} \in \mathbb{R}^k$ be defined as $$\mathbf{z} \colon= (1 - \alpha) \mathbf{x} + \alpha \mathbf{y}.$$ Then $$\vert \mathbf{z} - \mathbf{x} \vert = \alpha \vert \mathbf{x} - \mathbf{y} \vert = \alpha d, $$ and $$\vert \mathbf{z} - \mathbf{y} \vert = (1- \alpha) \vert \mathbf{x} - \mathbf{y} \vert = (1-\alpha )d. $$ So we have $$\vert \mathbf{z} - \mathbf{x} \vert = \vert \mathbf{z} - \mathbf{y} \vert = r$$ if and only if $$\alpha d = (1-\alpha)d = r,$$ which holds if and only if $$\alpha = \frac 1 2 \ \ \ \mbox{ and } \ \ \ r = \frac d 2.$$ So if $r = \frac d 2$, then the element $\mathbf{z} \in \mathbb{R}^k$ given by the formula $$ \mathbf{z} \colon= \frac 1 2 (\mathbf{x} + \mathbf{y})$ satisfies our relation.
Moreover, if $r = \frac d 2$ and if some element $\mathbf{z} \in \mathbb{R}^k$ satisfies our equality, then?
Also, an element $\mathbf{z} \in \mathbb{R}^k$ satisfies $$\vert \mathbf{z} - \mathbf{x} \vert = \vert \mathbf{z} - \mathbf{y} \vert = r$$ if and only if $$\vert \mathbf{z} - \mathbf{x} \vert^2 = \vert \mathbf{z} - \mathbf{y} \vert^2 = r^2,$$ and the last relation in terms of components becomes $$\sum_{j=1}^k (z_j- x_j)^2 = \sum_{j=1}^k (z_j - y_j)^2.$$ Hence $$\sum_{j=1}^k (y_j - x_j) ( 2 z_j - x_j - y_j) = 0.$$
How do we use the fact that $k \geq 3$?
And, how do we modify these statements if $k$ is $1$ or $2$?