Reflection $\mathscr{R}H$ through given hyperplane exchanges given points.

Question

Let $P$, $Q$ be two points in $\mathbb{R}^n$. Consider the hyperplane $H = \{v \in \mathbb{R}^n : \, |P − v| = |Q − v|\}$. Prove that the reflection through the hyperplane $H$, $\mathscr{R}H$, exchanges $P$ and $Q$.

Answer 1

You could just rotate and translate the entire situation to one you're familiar with. If you can prove the situation there and then rotate and translate back to your starting position, you're done.

The easiest way to do this, it seems, would be to first rotate your situation to one in which the line through $P$ and $Q$ is parrallel to the $x_1$-axis. You can then translate this situation to one where $P$ and $Q$ lie on the $x_1$-axis and have equal distances to the origin. At this point, your hyperplane is just a hyperplane through the origin, perpendicular to the $x_1$-axis and showing that $P$ and $Q$ are mapped onto one another shouldn't be too hard.