IMO 1969 LL CZS16

Origin: CZS

Problem

A convex quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d and angles \alpha = \angleDAB, \beta = \angleABC, \gamma = \angleBCD, and \delta = \angleCDA is given. Let s = (a + b + c + d)/2 and P be the area of the quadrilateral. Prove that P 2 = (s −a)(s −b)(s −c)(s −d) −abcd cos2 \alpha + \gamma .