Pythagorean Theorem in Euclidean Space

Let $n\in$ $\mathbb{N}$, and let $A$, $B$, $C$ be points of Euclidean space $\mathbb{R}^n$. Assume that the differences $B-A$ and $C-A$ are orthogonal, that is,

Then, with $d_E$ denoting the Euclidean distance on $\mathbb{R}^n$,

In the classical picture, $A$ is the vertex of the right angle of a right triangle, $d_E(A,B)$ and $d_E(A,C)$ are the lengths of the two legs, and $d_E(B,C)$ is the length of the hypotenuse.