Proving Inequalities for Cyclic Quadrilaterals

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Homework Help Overview

The problem involves proving an inequality related to a convex cyclic quadrilateral ABCD, specifically the relationship between the lengths of its sides and diagonals.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the nature of cyclic quadrilaterals, questioning the necessity of the convexity condition. There are mentions of using triangle inequalities related to the triangles formed by the quadrilateral, but attempts to manipulate these inequalities to match the problem statement have not progressed far.

Discussion Status

The discussion is ongoing, with participants exploring the properties of cyclic quadrilaterals and considering various inequalities. Some guidance has been offered regarding the triangle inequalities, but no consensus or clear direction has emerged yet.

Contextual Notes

There is a mention of the number of triangle inequalities that can be derived from the quadrilateral, indicating a potential complexity in the problem. The original poster's approach seems to be hindered by the challenge of relating these inequalities to the given statement.

ehrenfest
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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
 
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ehrenfest said:
First, isn't a cyclic quadrilateral always convex?

Hi ehrenfest! :smile:

Yes … "convex" seems unnecessary!
 
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.
 
tiny-tim said:
Hi ehrenfest! :smile:

Yes … "convex" seems unnecessary!

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.
 

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