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In triangle $A B C$, a rhombus $A D E F$ is inscribed such that angle $A$ is common to both, and vertex $E$ lies on side $B C$.

Find the side of the rhombus if $A B=c$ and $A C=b$.
The diagonal of a rhombus is the bisector of its angle.

## Solution

Let point $D$ lie on side $A B$ of triangle $A B C$. From the similarity of triangles $B D E$ and $B A C$, it follows that $D E : A C = B E : B C$. Since $A E$ is the bisector of angle $A$, $D E = A C \cdot \frac{B E}{B C} = \frac{b c}{b+c}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-33.jpg?height=348&width=606&top_left_y=2167&top_left_x=729)

## Answer

$\frac{b c}{b+c} \cdot$

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