The radius of a circle passing through a circle and the centre point of another smaller circle

Question

My idea was determining the equation of the smallest circle of which it might determine the equation of the moddle circle but I was not pretty sure

Answer 1

$$AC=2\sqrt{11^2-(11-8)^2}=2\sqrt{112}.$$ $$AB^2=\sqrt{112+4^2}=\sqrt{128}.$$ Thus, $$R=\frac{AB\cdot BC\cdot AC}{4S_{\Delta ABC}}=\frac{128\cdot2\sqrt{112}}{4\cdot\frac{4\cdot2\sqrt{112}}{2}}=16.$$