Question 115
Let $ABC$ be a triangle with $AB = 20$, $BC = 10$, $CA = 15$. Let $I$ be the incenter of $ABC$, and let $BI$ meet $AC$ at $E$ and $CI$ meet $AB$ at $F$. Suppose that the circumcircles of $BIF$ and $CIE$ meet at a point $D$ different from $I$. Find the length of the tangent from $A$ to the circumcircle of $DEF$.