Question

Abcd is a quadrilateral prove that ab+ cd + bc+ da is less than 2( ac+ bd )

Answer 1

Answer:

Let sides be a, b, c , d and diagonal p and q . we need to prove

a + b + c + d < 2 ( p + q )

Where p = p' + p" & q = q'+ q"

In a triangle sum of two sides is greater then third side

So in triangles

ABO , BCO , CDO & DAO

a < p' + q"

b < p" + q"

c < p" + q'

d < p' + q'

Add all equations

a + b + c + d <

p' + q" + p" + q" + p" + q' + p'+ q'

a + b + c + d < 2( p + q )