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$G$ is the centroid of $\triangle ABC.$ $G_1,G_2,$ and $G_3$ are the centroids of $\triangle BCG,\triangle CAG,$ and $\triangle ABG,$ respectively. What is $\dfrac{[G_1G_2G_3]}{[ABC]}?$ [asy] size(5cm); pair A,B,C,G,G1,G2,G3; A=(0,0); B=(1,3); C=(3,0); G=(A+B+C)/3; G1=(B+C+G)/3; G2=(A+C+G)/3; G3=(A+B+G)/3; draw(A--B--C--A--G--B^^G--C); draw(G1--G2--G3--G1); dot(Label("$A$", A, SW)); dot(Label("$B$", B, N)); dot(Label("$C$", C, SE)); dot(Label("$G$", G, S)); dot(Label("$G_1$", G1, N)); dot(Label("$G_2$", G2, E)); dot(Label("$G_3$", G3, N)); [/asy]

 Nov 28, 2021
 #1
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My friend, 

your laTex is quite unreadble and most probably not what you intended it to be. Please go on over to https://latex.codecogs.com/eqneditor/editor.php and re-write your question so I can help. 

 

Thanks. 

 Nov 28, 2021
 #2
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Using Barycentric coordinates, I found that $\frac{[G_1 G_2 G_3]}{[ABC]} = 1/12$.

 Nov 28, 2021

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