RO-NMO 2003 Problem 84

In tetrahedron $ABCD$ , $G_1,G_2$ and $G_3$ are barycenters of the faces $ACD,ABD$ and $BCD$ respectively. (a) Prove that the straight lines $BG_1,CG_2$ and $AG_3$ are concurrent. (b) Knowing that $AG_3=8,BG_1=12$ and $CG_2=20$ compute the maximum possible value of the volume of $ABCD$ .