TUYMAADA 2014 J6

Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$ . The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.

(L. Emelyanov)