(a) Given a general (not necessarily rectangular) tetrahedron, let $\displaystyle V_{1}, V_{2}, V_{3}, V_{4}$ denote vectors whose lengths are equal to the areas of the four faces, and whose directions are perpendicular to the these faces and point outward. Show that $\displaystyle V_{1} + V_{2} + V_{3} + V_{4} = 0$.

(b) Now consider a rectangular tetrahedron, i.e., one which has one vertex at which all three angles are right angles. Let D denote the area of the face opposite this rectangular vertex (i.e., the analog of a hypothenuse), and let A;B;C denote the areas of the other three faces. Prove that $\displaystyle D^2 = A^2 + B^2 + C^2$.