<Tri-2023-2027>, hint

Suppose the incircle touches side AB,AC at D,E respectively.

Then AX xAY=ADxAD, AX+AY>2root(AX xAY), and AD=AE=s-a, BD=BX=s-b, CX=CE=s-c

BC=s-b+s-c=2s-(b+c)=a.

AX+AY+BC>AB+AC is equivalent to 2root(AX xAY)+BC=2root (s-a)(s-a)+s-b+s-c=

(s-a)+(s-b)+(s-a)+(s-c)=AB+AC