All the proofs of the high-dimensional Pythagorean theorem that I know are based on induction or the additivity of the dot product. Is there any geometric construction that's similar to the well-known planar proofs, ones that are based on clever divisions of squares or similarities between certain triangles?



Geometric proof of the three-dimensional Pythagorean theorem
In a rectangular parallelepiped with edges $|AB|=a$, $|BC|=b$, $|CD|=c$ and space diagonal $|AD|=d$ let us draw altitudes $BX$ and $CY$ from the vertices to $AD$. Then using similarity arguments it is easy to see that $|AX|=a^2/d$ and $|YD|=c^2/d$. Using, say, a parallel translation and a similarity argument one also sees that $|XY|=b^2/d$. We obtain $a^2/d+b^2/d+c^2/d=d$. This is probably in some sense dual to what you wrote in your comment.


