Triangle $ABC$ has $AB=13$, $BC=14$ and $AC=15$. Let $P$ be the foot of the altitude from $B$ to $\overline{AC}$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ (with $AD\parallel BC$) and $ABCE$ (with $CE\parallel AB$) are trapezoids. What is the distance $DE$?