Let $ABCD$ be a convex quadrilateral and let $O$ be the point of intersection of its diagonals. Prove that if the perimeters of $\triangle ABO$,$\triangle BCO$,$\triangle CDO$ and $\triangle DAO$ are equal then $ABCD$ is a rhombus.
first time I saw this problem on Problem-Solving Strategies book By Arthur Engel on summer(you can see this solution from this book on page 126).this problem was in the extremal principle chapter.I wanted to find another way to prove it but I was not able to do anything useful.
I did some research and I was able to find that this problem is From MOSP 2001. But I was not able to find any other solution other than the one I saw on Arthur Engel book. Any help or hint on proving this problem in other ways are appreciated.