I found a problem in Arnold's Mathematical Methods of Classical Mechanics Chapter 28 as follows:
Draw the line through the center of a cube such that the sum of the squares of its distances to the vertices of the cube is (a) largest, (b) smallest.
I found by example that the sum of squares should be a constant which is independent of the line. In the two dimensional analogue there is an easy geometric proof using Pythagorean theorem. My question is: how to prove it for a cube?