Hello, mathwizard325!

I've proved it one way so far . . . a vector proof.

Prove that if two diagonals of a rectangular parallelepiped are perpendicular,

then its dimensions are equivalent to the sides of a right triangle and vice versa.

The "box" has dimensions: . in the direction,

. . in the direction, in the direction.

Orient it in the first octant with vertex at the origin.

The graph looks like this:

Code:

T | W
(0,0,c)* - - - - - * (0,b,c)
/| /|
/ | / |
U / | / |
(a,0,c) * - - - - - * V |
| | | | S
| P * - - - | - * (0,b,0)
| / | /
| / | /
Q |/ |/
(a,0,0) * - - - - - * R
/

One diagonal is: .

. . . . Another is: .

Since

Therefore: .