MathDB

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.

Problem Statement