The problem in the image describes a geometric configuration with triangles $ABC$, $ADF$, and $DBE$, where specific relationships between angles and sides are outlined, particularly focusing on the areas of triangles $ADF$ and $DBE$ as $a$ and $5a$, respectively. There is a detailed explanation of the similarity of triangles based on angle relationships and proportionality (notably, using angle-angle similarity and the concept of side proportionality $k$).

Key points from the image:

1. The triangles $ADF$ and $DBE$ are similar due to angle congruence (marked with angle symbols), implying that corresponding sides of these triangles are proportional by a ratio $k$, where $k^2 = \frac{a}{5a} = \frac{1}{5}$, leading to $k = \frac{1}{\sqrt{5}}$.

2. The equations to calculate the