As title suggests, the question is to find the area of the shaded quadrilateral inside a triangle formed via two intersecting cevians, given the area of $3$ smaller triangles with areas $3$, $6$ and $9$ respectively.
I discovered this puzzle online posted on a language-learning app, and I found it quite interesting. The asker claimed that it was a grade 9 problem. I did attempt it in several ways, but a lot of the attempts did not lead anywhere. I will post my successful attempt as an answer below, please do let me know if it is correct, if something can be improved or if the answer may be wrong (the asker did not reveal the answer). And please share your own attempts too!
Best Answer
Let us denote by $X$ the area to be computed, and let us use letters for the relevant vertices as in the following picture:
Then applying Menelaus in $\Delta EFB$ w.r.t. the line $CDA$ we get: $$ 1= \frac{CE}{CF}\cdot \frac{DF}{DB}\cdot \frac{AB}{AE} = \frac{6+9}{9}\cdot \frac{3}{3+9}\cdot \frac{X+3+6+9}{X+3}\ , $$ which gives a first order equation in $X$ with unique solution $X=54/7$.