\left.\begin{array}{l}{[\text { Relations of linear elements of similar triangles }]} \\ {[\quad \text { Properties and characteristics of an isosceles triangle. }]}\end{array}\right]

In triangle $ABC$, side $AC$ is equal to $b$, side $AB$ is equal to $c$, $AD$ is the angle bisector, and $DA=DB$. Find the length of side $BC$.
Triangles $D A C$ and $A B C$ are similar. The ratio of the sum of sides $D A$ and $D C$ of triangle $A D C$ to the sum of sides $A B$ and $A C$ is equal to the similarity coefficient.

## Solution

Since triangle $A D B$ is isosceles, then $\angle B=\angle B A D=\angle D A C$. Therefore, triangles $A D C$ and $B A C$ are similar by two angles. Thus, $\frac{b}{B C}=\frac{A D+D C}{A C+A B}=\frac{B D+D C}{A C+A B}=\frac{B C}{b+c}$. From this, $B C^{2}=b(b+c)$.

## Answer

$\sqrt{b(b+c)}$.