Question 7
Triangle $ABC$ has $AB = 21,$ $AC = 22,$ and $BC = 20.$ Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC},$ respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC.$ Then $DE = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$