E 24. Egyptian Area Formula
Let ABCD be a quadrilateral with area K and let w = AB, x = BC, y = CD, z = DA. The ancient Egyptians used the expression K' = [(w + y)/2][(x + z)/2] for the area. Since they used this formula to compute the area of fields for taxing purposes, the government probably didn't mind that this formula overestimates the area for all quadrilaterals except rectangles. Prove this fact, that is K <= K', with K = K' iff ABCD is a rectangle. The proof is surprisingly easy, and only requires high school mathematics.