MathNet 2014 Question 36

Let $ABC$ be a triangle with $AB = AC = 5$ and $BC = 6$. Denote by $\omega$ the circumcircle of $ABC$. We draw a circle $\Omega$ which is externally tangent to $\omega$ as well as to the lines $AB$ and $AC$ (such a circle is called an $A$-mixtilinear excircle). Find the radius of $\Omega$.



Supplemental illustration of the given configuration.