If O is a point within a quadrilateral ABCD,


If O is a point within a quadrilateral ABCD, show that OA + OB + OC + OD > AC + BD.



Let ABCD be a quadrilateral whose diagonals are AC and BD and O is any point within the quadrilateral. 
Join O with A, B, C, and D.
We know that the sum of any two sides of a triangle is greater than the third side.
So, in ∆AOCOA + OC > AC

Also, in ∆ BODOB + OD > BD
Adding these inequalities, we get:
(OA + OC) + (OB + OD) > (AC + BD)
⇒ OA + OB + OC + OD > AC + BD

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