Task 1451

The task

Let $ABCD$ be a convex quadrilateral, and let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively. Find $\frac{[G_A G_B G_C G_D]}{[ABCD]}.$ [asy] unitsize(0.6 cm); pair A, B, C, D; pair[] G; A = (0,0); B = (7,1); C = (5,-5); D = (1,-3); G[1] = (B + C + D)/3; G[2] = (A + C + D)/3; G[3] = (A + B + D)/3; G[4] = (A + B + C)/3; draw(A--B--C--D--cycle); draw(G[1]--G[2]--G[3]--G[4]--cycle,red); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, SE); label("$D$", D, SW); dot("$G_A$", G[1], SE); dot("$G_B$", G[2], W); dot("$G_C$", G[3], NW); dot("$G_D$", G[4], NE); [/asy]