MathNet 2015-part-07 — 185 problems (EN–VI) Question 83

Let $ABC$ be a triangle whose incircle has center $I$ and is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$, at $D, E, F$. Denote by $X$ the midpoint of major arc $\widehat{BAC}$ of the circumcircle of $ABC$. Suppose $P$ is a point on line $XI$ such that $\overline{DP} \perp \overline{EF}$.
Given that $AB=14$, $AC=15$, and $BC=13$, compute $DP$.