let $ABC$ be a triangle with $[ABC]=S_{ABC}=16$ and $D,E,F$ point in $AB,BC,AC$ respectively , such that :
$AB=3AD$ , $CA=3CF$ , $BC=3BE$
find the area of triangle $DEF$ , $S_{DEF}=$ ?
Actually I can't starte to solve this ?
My be we can solve it by barycenter coordinate but I don't understand why
$[DEF]=[ABC]\begin{vmatrix} x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{vmatrix}$
Where $x_{i}$ coordinate of barycenter I don't understand why this relations I need simple prof ?
I'm searching a esay method to find area ?
Any other ways ?
The area of $DEF$ is $16/3$ for the following reason.
Connect $D$ to $G$ where $G$ is the midpoint of $AF$
Note that the area of $ADG$ is $1/9$ of the area of $ABC$
That makes the area of $ADF$ to be $2/9$ of the area of $ABC$
Similarly you have the area of each one of the other two outer triangles to be $2/9$ of the area of $ABC$ which leave the middle area to be $1/3$ of the area of $ABC$.
Thus the area of $DEF$ is $16/3$
Similar argument shows that if you change the integer $3$ to $k$ the ratio of the area of the middle triangle to the original triangle will be $$\frac {k^2-3k+3}{k^2}$$