<Tri-2025-225>, hint

Suppose circle S(a) touches BC at D, and R is the radius of circumcircle of ABC 

and angle A' is the angle of S(a), between the two tangents from A to S(a)

Since OD=R cos A,  sine( A'/2)=R cosA/R=cosA=sin(90-A)

Hence A'=180-2A, similarly B'=180-2B, C'=180-2C.

Hence A'+B'+C'=540-2(A+B+C)=180