Given two points $x, y \in \mathbb{R}^k, k\geq 3$ and one scalar $r \in \mathbb{R}$, the goal is to prove that there are an infinite amount of points $z \in \mathbb{R}^k$ such that: $$|z - x| = |z - y| = r$$
Graphically, all vectors $z$ should form a hyperplane sitting in between $x$ and $y$.
I couldn't find a way to algebraically manipulate the equation, and I'm pretty sure that operating on the components is out of the question.
My next approach was constructing a vector which satisfied the equation, and then show that scaling it would not affect the result. Intuitively, this would be a vector $b$ such that:
$$b \cdot (x - y) = 0$$
eminating from the midpoint between $x$ and $y$:
$$z = \frac{1}{2}(x + y) + \alpha b, \alpha \in \mathbb{R}$$
I think this defines an infinite subset of all $z$ values, but I've had no luck trying to plug it back into the equation.