Two much of a good thing?

$\triangle ADF$ and $\triangle ABE$ have congruent incircle and three vertex angles. The two triangles are congruent and therefore $DF=AB=6$ .

Extend $DC$ and $AE$ to meet at $G$ . We note that $\triangle DFG$ and $\triangle ABE$ are similar. Then the ratio of the radius of the blue circle to that of the red circle $\dfrac {r_\blue{\rm blue}}{r_\red{\rm red}} = \dfrac {DF}{EB} = \dfrac 6{\sqrt{8^2-6^2}} = \dfrac 6{2\sqrt 7} = \dfrac 3{\sqrt 7}$ . Therefore $a+b = 3+7 = \boxed{10}$ .