<Tri-2025-2221>, hint

Let AM/MB=BN/NC=CP/PA=t/1-t and A(x',y'), B(x",y"), C(x"', y"')

Then M=tx"+(1-t)x', N=tx"'+(1-t)x", P=tx'+(1-t)x"'

G(1)=1/3(A+M+P)=1/3{x'+tx"+(1-t)x'+tx'+(1-t)x"'}, Similarly G(2), G(3) defined

You can find G=1/3 (A+B+C)=1/3{G(1)+G(2)+G(3)}