Olympiad Geometry · Superchallenge problem

Triangle $ABC$ has $AB = AC = 13$ and $BC = 10$. Point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ with $\overline{DE} \parallel \overline{BC}$, and segment $\overline{DE}$ cuts the triangle into a smaller triangle and a trapezoid whose perimeters are equal. Find $DE$.