Suppose k\(\displaystyle \ge 3, x, y \in \mathbb{R}^k, ||x-y|| = d > 0\), and \(\displaystyle r > 0\). Prove (a) that if \(\displaystyle 2r > d\), there are infinitely many \(\displaystyle z \in \mathbb{R}^k\) such that \(\displaystyle ||z - x|| = ||z - y|| = r\), (b) there is exactly one such \(\displaystyle z\) if \(\displaystyle 2r = d\) and (c) there is no such z if \(\displaystyle 2r < d\).

My progress:

I proved (b) and (c) using the condition for attainment in Cauchy-Schwarz. Part (a) I proved after assuming that \(\displaystyle x = 0\) and \(\displaystyle y = (y_1, 0, ..., 0)\). I'm probably just being stupid, but the problem I seem to be running into is that I'm having a hard time undoing my rotation and translation without going beyond what can be done with the material learned so far. Any help would be appreciated. If anyone could give a simple justification (i.e. one that doesn't use anything but definition of the Euclidean norm and the basic algebraic properties of the dot product) for assuming WLOG that \(\displaystyle x = 0\) and \(\displaystyle y = (y_1, 0, ..., 0)\), this would certainly do the trick. Any other permissible solution would be fine too.

Thanks.