This problem is from V. Prasolov: Problems in plane and solid geometry. For those interested in solution I have attached one solution that I have extracted from Croatian 2010. booklet.

Attachments:
Bosnian TST 2010 problem 4.pdf (205kb)

Let the diagonals intersect at $S$. Not hard to verify that $S$ must be the excircle touchpoint of all triangles $\triangle ABC,\triangle BCD,\triangle CDA,\triangle DAB$ (by Casey's theorem). Then since $S$ divides these segments in $s-a,a$ where $s$ is the semi perimeter, we must have that all sides are equal by writing out the simple equations.