4B. The tangent to the inscribed circle of triangle $A B C$, parallel to side $A B$, intersects sides $A C$ and $B C$ at points $A_{1}$ and $B_{1}$, respectively. Calculate the length of segment $A_{1} B_{1}$, if the lengths of the sides of triangle $A B C$ are $a, b$, and $c$.
Solution. Let $\overline{A_{1} C}=b_{1}, \overline{B_{1} C}=a_{1}, \overline{A_{1} B_{1}}=x$. From the similarity of triangles $A B C$ and $A_{1} B_{1} C$, it follows that $\overline{A B}: \overline{B C}=\overline{A_{1} B_{1}}: \overline{B_{1} C}$, that is,

$$
a_{1}=\frac{a}{c} x
$$

From the same similarity, it follows that $\overline{A B}: \overline{A C}=\overline{A_{1} B_{1}}: \overline{A_{1} C}$, that is,

$$
b_{1}=\frac{b}{c} x
$$

Since quadrilateral $A B A_{1} B_{1}$ is tangential, it follows:

![](https://cdn.mathpix.com/cropped/2024_06_05_5122104380b1afd5c770g-03.jpg?height=375&width=394&top_left_y=634&top_left_x=1067)

$$
\overline{A B}+\overline{A_{1} B_{1}}=\overline{A A_{1}}+\overline{B B_{1}}
$$

that is,

$$
c+x=b-b_{1}+a-a_{1}
$$

From the last equality and equalities (1) and (2), it follows that $x=\frac{a+b-c}{a+b+c} c$.

## II year