3. In $\triangle A B C$, $3 B C=A B+C A, T$ is a point on side $A C$ such that $4 A T=A C$, points $K$ and $L$ are on sides $A B$ and $C A$ respectively, satisfying $K L / / B C$, and line $K L$ is tangent to the incircle of $\triangle A B C$. Find the ratio of the two segments into which $B T$ divides $K L$.


3. As shown in Figure 2, in $\triangle A B C$, let the altitude from point $A$ to the opposite side be $h$, and the radius of the incircle of $\triangle A B C$ be $r$.
From the area formula of $\triangle A B C$, we know
$$
r \frac{A B+B C+C A}{2}=\frac{B C h}{2} \Rightarrow h=4 r \text {. }
$$

Since $K L / / B C$ and the distance between the two lines is $2 r$, in $\triangle A K L$, the altitude from point $A$ to the opposite side is $2 r$, and $\triangle A K L \backsim \triangle A B C$.
Thus, $K$ and $L$ are the midpoints of $A B$ and $A C$ respectively
$\Rightarrow T$ is the midpoint of $A L$
$\Rightarrow B T$ and $K L$ are the medians of $\triangle A B L$.
Let $B T$ and $K L$ intersect at point $G$.
Therefore, $\frac{K G}{G L}=\frac{1}{2}$.