Junior Macedonian Mathematical Olympiad

Let $AC\cap BD=O$. Clearly, $AO$ and $BH$ are medians in the triangle $ABD$, hence $I$ is the centroid of $ABD$. Similarly $K$ is the centroid of $BCD$. If $\overline{IO}=x$, then $\overline{AI}=2x$. Similarly, if $\overline{KO}=y$, then $\overline{CK}=2y$. Therefore $3x=\overline{AO}=\overline{CO}=3y$, i.e. $x=y$. We analogously prove that $\overline{JO}=\overline{LO}$. It follows that $IJKL$ is a parallelogram.