Question

If a rectangular box is inscribed in a sphere, can we assume that the space diagonal of the box is indeed the diameter of the sphere? And if so, is there a theorem or proof for this?

Answer 1

You could use the fact that the diagonal of a rectangular box defines a right triangle with vertices on the surface of the sphere. Recall that if a right triangle has its vertices on a circumference, it's hypotenuse is a diameter of the circumference. Then you could try and expand it to a sphere, but you'd have to guarantee that this right triangle lies on a plane that passes through the center of the sphere.