[ Ratio of areas of similar triangles ]

A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2: 1$, counting from the vertex. In what ratio does it divide the lateral sides?
The given straight line cuts off a triangle similar to the given triangle. The similarity coefficient is $\frac{\sqrt{2}}{\sqrt{3}}$. Therefore, the side of the triangle is divided by this line in the ratio

$\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\sqrt{6}+2$, counting from the vertex.

## Answer

$(\sqrt{6}+2): 1$