Question 116
Let $ABC$ be a triangle with $BC = 2007$, $CA = 2008$, $AB = 2009$. Let $\omega$ be an excircle of $ABC$ that touches the line segment $BC$ at $D$, and touches extensions of lines $AC$ and $AB$ at $E$ and $F$, respectively (so that $C$ lies on segment $AE$ and $B$ lies on segment $AF$). Let $O$ be the center of $\omega$. Let $\ell$ be the line through $O$ perpendicular to $AD$. Let $\ell$ meet line $EF$ at $G$. Compute the length $DG$.