Question 10
$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}$, $\overline{C A}$, and $\overline{A B}$ at $D$, $E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\overline{A C}$ and $\overline{A B}$ at $X$ and $Y$ respectively. Given $X Y = 8$, $A E = 15$, and that the radius of $\Gamma$ is $5$, compute $B D \cdot D C$.