[ Properties of medians. The centroid of a triangle.] [ The ratio of the areas of similar triangles $]$

The area of triangle $A B C$ is $S$. Find the area of the triangle whose sides are equal to the medians of triangle $A B C$.
Let $M$ be the point of intersection of the medians of triangle $ABC$, $B_{1}$ be the midpoint of side $AC$. Extend median $BB_{1}$ beyond point $B_{1}$ and lay off segment $B_{1}K = B_{1}M$. Since $AMCK$ is a parallelogram, $KC = AM$. Therefore, the sides of triangle $MCK$ are equal to $2/3$ of the sides of the triangle formed by the medians of triangle $ABC$. Hence,

![](https://cdn.mathpix.com/cropped/2024_05_06_5e9166ba6ad1f1b1f21fg-49.jpg?height=452&width=423&top_left_y=154&top_left_x=818)

## Answer