Question 95
In triangle $ABC$, a circle $\omega$ with center $O$ passes through $B$ and $C$ and intersects segments $\overline{AB}$ and $\overline{AC}$ again at $B'$ and $C'$, respectively. Suppose that the circles with diameters $BB'$ and $CC'$ are externally tangent to each other at $T$. If $AB = 18$, $AC = 36$, and $AT = 12$, compute $AO$.