Question 67
Let $ABC$ be a triangle with $AB=13$, $BC=14$, $CA=15$. The altitude from $A$ intersects $BC$ at $D$. Let $\omega_{1}$ and $\omega_{2}$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_{1}$ and $\omega_{2}$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.