# Is the sum of the diagonals always greater than the sum of opposite sides in a convex quadrilateral?

Given a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ then lie inside $ABCD$. Then must it be true that $AC+BD>AB+CD$ and $AC+BD>AD+BC$?

I have been unable to come up with a counterexample to this, and have also made no progress trying to prove it. Any suggestions would be appreciated.

## 1 Answer

Label the intersection between $AC$ and $BD$ $O$, then use Triangle Inequality multiple times.