Answer: 378

Small Hint:

Find $\omega_A$ first: its center lies on line $AO,$ its radius is half its distance from $A,$ and internal tangency to $\omega$ fixes everything

Big Hint:

The two intersection points of $\omega_B$ and $\omega_C$ lie on line $AO,$ and each triangle’s circumradius is the distance from that point to the center $O$

Solution:

Let $O$ be the center of $\omega.$ The center of $\omega_A$ lies on line $AO$ (the bisector of $\angle A$) at some distance $d$ from $A;$ since $\overline{AB}$ makes a $30^\circ$ angle with $AO,$ the radius is $r = d \sin 30^\circ = \frac{d}{2}.$ Internal tangency to $\omega$ requires the center to be $18 - r$ from $O,$ which forces the center past $O:$ $d - 18 = 18 - \frac{d}{2},$ so $d = 24,$ $r = 12,$ and the center is $6$ beyond $O.$

Place $O$ at the origin with $A = (0, 18).$ Then the three centers are $O_A = (0, -6)$ and $O_B, O_C = (\pm 3\sqrt{3}, 3),$ all with radius $12.$ The intersections of $\omega_B$ and $\omega_C$ lie on the $y$-axis: $27 + (y - 3)^2 = 144$ gives $y = 3 \pm \sqrt{117}.$ The point $(0, 3 + \sqrt{117})$ is closer to $A$ and belongs to the larger triangle, so the smaller triangle has vertex $(0, 3 - \sqrt{117}),$ at distance $\sqrt{117} - 3$ from $O.$

By symmetry the smaller triangle is equilateral with circumradius $\sqrt{117} - 3,$ so its side is $\sqrt{3}\left(\sqrt{117} - 3\right) = \sqrt{351} - \sqrt{27}.$ Thus $a + b = 351 + 27 = 378.$