VWO 1992 Problem 4

Let $A,B,P$ positive reals with $P\le A+B$ . (a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that $$A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)}$$ (b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$ . (c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$ , $z,w>0$ and $z^2+w^2<x^2+y^2$ . Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$ , the cyclical one has the greatest area.