<Tri-2025-2282>, hint

Let I be incenter of ABC, lines DC' and DB' cut w at C", B". 

Let<BAA'=X, CAA'=Y, Since AC'=AD=AB'.<AC'D=X/2, <AB'D=Y/2 ,<A=X+Y

Hence <C'IC"=X ,<B'IB"=Y. Hence <B"IC"=<B'IC'+<B'IB"+<C'IC"=180-(X+Y)=X+Y=180

Thus B"IC" is diameter of circle w.