Two triangles and their content ratio

Question

I would like to ask if someone could help me with solving the following task.

Find the area ratio between ABC and the hatched triangle.

Thank you in advance.

Answer 1

COMMENT: If you do not want to use Routh's theorem you may reason as follows:

Considering this sketch it can be shown that:

$BP=PN=3NC_1$ ⇒ $NC_1= \frac{ BC_1}{7}$

Similarly:

$MA_1=\frac{CA_1}{7}$

$PB_1=\frac{AB_1}{7}$

Also:

$S_{\triangle BCC_1}=S_{\triangle CAA_1}=S_{\triangle ABB_1}=\frac{S_{\triangle ABC}}{3}$

Each two of these three triangles have common triangles:

$\triangle BCC_1$ and $\triangle CAA_1$ have $\triangle NCC_1$

$\triangle BCC_1$ and $\triangle ABB_1$ have $\triangle PBB_1$

$\triangle ABB_1$ and $\triangle CAA_1$ have $\triangle MAA_1$

The sum of areas of these three triangle is equal area of triangle MNP. Also:

$S_{MAA_1}=S_{PBB_1}=S_{NCC_1}=\frac {S_{ABB_1}}{7}$

⇒$S_{MNP}=\frac{3\times S_{ABB_1}}{7}=\frac{S_{ABC}}{7} $