On the extensions of the medians $A K, B L$, and $C M$ of triangle $A B C$, points $P, Q$, and $R$ are taken such that $K P=\frac{1}{2} A K, L Q=\frac{1}{2} B L$, and $M R=\frac{1}{2} C M$. Find the area of triangle $P Q R$, if the area of triangle $A B C$ is 1.
Find the similarity coefficient of triangles $K M L$ and $P Q R$.

## Solution

Let $O$ be the point of intersection of the medians of triangle $A B C$. Triangle $K M L$ is similar to triangle $A C B$ with a coefficient of $\frac{1}{2}$, and triangle $P Q R$ is similar to triangle $K L M$ with a coefficient of

$$
\frac{O P}{O K}=\frac{O K+K P}{O K}=\frac{\frac{1}{3} A K+\frac{1}{2} A K}{\frac{1}{3} A K}=\frac{5}{2}
$$

Therefore,

$$
S_{\triangle \mathrm{PQR}}=\frac{25}{4} S_{\Delta \mathrm{KLM}}=\frac{25}{4} \cdot \frac{1}{4} S_{\Delta \mathrm{ABC}}=\frac{25}{16}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-52.jpg?height=417&width=826&top_left_y=1221&top_left_x=619)

Answer

$\frac{25}{16}$.