Putnam and Beyond prob 117

[ehrenfest]

Homework Statement

Let ABCD be a convex cyclic quadrilateral. Prove that

$$|AB-CD|+|AD-BC| \geq 2|AC-BD|$$

Homework Equations

The Attempt at a Solution

First, isn't a cyclic quadrilateral always convex?

http://en.wikipedia.org/wiki/Cyclic_quadrilateral

 

[tiny-tim]

First, isn't a cyclic quadrilateral always convex?

Hi ehrenfest!

Yes … "convex" seems unnecessary! :yuck:

 

[DavidWhitbeck]

Putnam is supposed to be for fun Ehrenfest. If you ask for help on every problem that you can't immediately solve... how are you having fun? The pleasure is all in finding the aha! moment yourself.

 

[ehrenfest]

Hi ehrenfest!

Yes … "convex" seems unnecessary! :yuck:

So, I can get triangle inequalities for the triangles ABC, BCD, ACD, ABD. Put there are 12 of them and I tried to play around with so they would look similar to the inequality in the problem statement but I did not get very far.