In the above diagram, $\triangle ABC$ is an equilateral triangle and point $O$ is the center of the circle inscribed in $\triangle ABC.$ If the length of $\overline{AB}$ is $\lvert \overline{AB} \rvert = 27,$ $\overline{AB} \text{ and } \overline{OD}$ are parallel, and $\overline{AC} \text{ and } \overline{OE}$ are parallel, what is $\lvert\overline{DE}\rvert(=x)?$