Question 84
Let $ABC$ be a triangle with $AB = 5$, $BC = 8$, and $CA = 7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $ABC$ at $A$ which is also tangent to segment $BC$. $\Gamma$ intersects $AB$ and $AC$ at points $D$ and $E$, respectively. Determine the length of segment $DE$.