Question

How to do inequalities with respect to sides of a quadrilateral?

Answer 1

question:
Let  ABCD be a quadrilateral with [ABCD] =1. Let s be semi-perimeter; p,q lengths of the diagonals, respecitively. Prove that:s+(p+q)/2 ≥2+√2.
sol:

Let ,AB=a, BC=b, CD=c and DA=d.

Hence,

1= SABCD ≤ SΔABC+SΔACD≤1(ab+cd)/2.

By the same way we'll obtain 1≤12(ad+bc)1≤12(ad+bc). Thus, (a+c)(b+d)=ab+cd+ad+bc≥4(a+c)(b+d)=ab+cd+ad+bc≥4.

In another hand, 1=SABCD≤12pq1=SABCD≤12pq, which gives pq≥2pq≥2.

Id est, by AM-GM

s+(p+q)/2= (a+b+c+d)/2+(p+q)/2 ≥ √(a+c)(b+d)+√pq≥2+√2Done!