$$
\text { [ Rhombi. Properties and Characteristics ] }
$$

A rhombus is inscribed in a triangle such that one angle is common to both, and the opposite vertex divides the side of the triangle in the ratio $2: 3$. The diagonals of the rhombus are $m$ and $n$. Find the sides of the triangle that contain the sides of the rhombus.

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Two sides of the rhombus cut off from a given triangle similar triangles.

## Solution

Let the vertices $M, K$ and $N$ of the rhombus $A M K N$ be located on the sides $A B, B C$ and $A C$ of triangle $A B C$, respectively. The side of the rhombus is equal to $1 / 2 \sqrt{m^{2}+n^{2}}$.

From the similarity of triangles $C K N$ and $C B A$, we find that $N K: A B = C K: C B = 3: 5$. Therefore, $A B = 5 / 3 N K$. Similarly, we find the side $A C$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-34.jpg?height=283&width=598&top_left_y=2214&top_left_x=730)

## Answer

$5 / 6 \sqrt{m^{2}+n^{2}}, 5 / 4 \sqrt{m^{2}+n^{2}}$