3. Let $E$ be a point on the side $AB$ of square $ABCD$ such that $|AE|=3|EB|$, and let $F$ be a point on the side $DA$ such that $|AF|=5|FD|$. Denote the intersection of segments $DE$ and $FC$ by $K$, the intersection of $DE$ and $BF$ by $L$, and the intersection of $FB$ and $EC$ by $M$. Let $p_{1}$ be the sum of the areas of triangles $EML$ and $DKC$, and $p_{2}$ be the sum of the areas of triangles $FLK$ and $MBC$. Determine the ratio $p_{1}: p_{2}$.
## III/3. Let $p$ be the area of quadrilateral $C K L M$. Then

 $p_{2}+p$ is equal to the area of triangle $F B C$. The length of the height to side $B C$ in triangle $F B C$ is equal to $|A B|$, so the area of the triangle is $\frac{|A B| \cdot|B C|}{2}=\frac{|A B|^{2}}{2}$. Therefore, 
$$
p_{2}=\frac{|A B|^{2}}{2}-p
$$

Similarly, we conclude that $p_{1}+p$ is equal to the area of triangle $D E C$, which is $\frac{|D C| \cdot|C B|}{2}=\frac{|A B|^{2}}{2}$. Thus, we also have

![](https://cdn.mathpix.com/cropped/2024_06_07_a8720c28cfa2c07912d2g-11.jpg?height=483&width=491&top_left_y=1666&top_left_x=1362)

$$
p_{1}=\frac{|A B|^{2}}{2}-p
$$

and the ratio $p_{1}: p_{2}$ is equal to 1.
Introduction of the area of quadrilateral $C K L M$ ..... 3 points
Writing $p_{1}=\frac{|A B|^{2}}{2}-p$ (or similar) ..... 1 point
Writing $p_{2}=\frac{|A B|^{2}}{2}-p$ (or similar) ..... 1 point
Conclusion $p_{1}: p_{2}=1$ ..... 2 points