Question: the egyptians thought that if a quadrilateral had sides of...
The Egyptians thought that if a quadrilateral had sides of
lengths a, b,c, and d, then its area S was given by the formula
(a+c)(b+d)/4. Prove that actually
4S is less than or equal to (a+c)(b+d)
with equality holding only for rectangles. (Hint:Twice the area of a triangle is ab sin θ, where θ is the angle between the sides of length a,b, and sinθ is less than or equal to 1, with equality holding only if θ is a right angle.)
Please prove this for when a quadrilateral is NOT CONVEX. Image below.