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Points $D,E,F$ are the first trisection points of $BC,CA,AB$ respectively. Let $[ABC]$ denotes the area of triangle $ABC$. If $[ABC]=1$, find $[GHI]$, the area of the shaded triangle. (the two images are from "The Art and Craft of Problem Solving" -Paul Zeitz)

attempt:

Let us look at the image below first

We know $AF:FB = 1:2$ ,this implies $[FIA]=x \implies [FBI]=2x$. Similarly $[DIB]=2y \implies [CID] = 4y$

Because $[CAF]:[CFB]=1:2$ and $[CFB]=6y+2x$, then $[CAF]=3y+x$ and $[CIA]=3y$.

We know the ratio $AF:AB = 1:3$, so ratio $[CAF]:[CFB]=1:2$.

Why at some point we can conclude that each of the three cevians $AD, EB, CF$, are divided by the ratio $1:3:3$?