3 [ Properties and characteristics of an isosceles triangle.]

In triangle $A B C$, angle $B$ is $90^{\circ}, A B=B C=2$. Points $K$ and $L$ are taken on the base $A C$ such that the three angles between $B A$ and $B K, B K$ and $B L, B L$ and $B C$ are equal to each other. Find the length of the segment $B K$.
Let point $K$ lie between $A$ and $L$. Drop a perpendicular $K N$ to $A B$. Let $B K=2 x$, then from triangle $B K N$ we get $K N=x, B N=x \sqrt{3}$. Moreover, $A N=K N$. Therefore,

$2=A B=x(1+\sqrt{3}), \quad B K=2 x=\frac{4}{\sqrt{3}+1}=2(\sqrt{3}-1)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_acc13965d0370bfc03dcg-37.jpg?height=372&width=684&top_left_y=1652&top_left_x=687)

## Answer

$2(\sqrt{3}-1)$.