<Tri-2025-2270>, hint

Let T be the point of tangency    at which circles w touches the circumcircle of ABC.

OT = R(circumradius), O; circumcenter of ABC

CA'=s-c thus A(0)A'=a/2-(s-c)=(c-b)/2=R(sinC-sinB)

OT-OA(0)=A(0)T--> R-R cosA=R(sinC-sinB) -->R(1-cosA)=R(sinC-sinB)

2sin^2(A/2)=2cos(C+B)/2 sin(C-B)/2--->sinA/2=sin(C-B)/2--->A+B=C, <C=90

a-r+b-r=c 2r=a+b-c ,OT-OA(0)=A(0)T--->c/2-b/2=a/2-r..> 2r=a+b-c

Triangle ABC, <C=90, Similarly in case of AC.