Consider a linear transformation $A: \mathbb{R}^{m} \longrightarrow \mathbb{R}^{n}$. I have to prove equivalence of 6 claims. I I couldn't prove one implication.
$$|Ax - Ay| = |x - y| \Longrightarrow \langle Ax, Ay \rangle = \langle x, y \rangle$$
Any hint? I didn't want the solution.
Note that from the assumptions, $$|Ax| = |Ax - A0| = |x - 0| = |x|.$$ Fix $x, y \in \mathbb{R}^m$. Then \begin{align*} &|x - y|^2 = |Ax - Ay|^2 \\ \implies \, &|x|^2 + |y|^2 - 2\langle x, y\rangle = |Ax|^2 + |Ay|^2 - 2\langle Ax, Ay\rangle \\ \implies \, &2\langle x, y\rangle = 2\langle Ax, Ay\rangle. \end{align*}