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Area of quadrilateral

Source:

9/29/2010
(CZS5)(CZS 5) A convex quadrilateral ABCDABCD with sides AB=a,BC=b,CD=c,DA=dAB = a, BC = b, CD = c, DA = d and angles α=∠DAB,β=∠ABC,γ=∠BCD,\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD, and δ=∠CDA\delta = \angle CDA is given. Let s=a+b+c+d2s = \frac{a + b + c +d}{2} and PP be the area of the quadrilateral. Prove that P2=(s−a)(s−b)(s−c)(s−d)−abcdcos⁡2α+γ2P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}
geometrytrigonometryconvex quadrilateralTrigonometric IdentitiesIMO ShortlistIMO Longlist