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: .