Area of the grey region in the following triangle > $ABC$ is a triangle having area of $240$. > > Given that $BD:DF:FC=3:4:3$; $BE:EA=5:7$ ; $CG:GA=2:7$ then what is the area of the grey region? >

Area of the grey region in the following triangle > $ABC$ is a triangle having area of $240$. > > Given that $BD:DF:FC=3:4:3$; $BE:EA=5:7$ ; $CG:GA=2:7$ then what is the area of the grey region? > > ![ABC is a triangle having area of 240 ]( I am not getting any way to start this question, any hint on the theorem will be helpful.

$$\frac{S_\Delta{BED}}{S_\Delta ABC}=\frac{5\cdot3}{12\cdot10}=\frac{1}{8};$$

$$\frac{S_\Delta{GFC}}{S_\Delta ABC}=\frac{2\cdot3}{9\cdot10}=\frac{1}{15}$$ Thus, $$S_{AEDFG}=240\left(1-\frac{1}{8}-\frac{1}{15}\right)=194.$$