An Olympiad Problem

Geometry Level 5

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 \(DE=\dfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

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