Problem 99. In triangle $ABC$ with sides $a, b, c$, lines are drawn through the incenter parallel to the sides of the triangle. Find the lengths of the segments of these lines that are intercepted by the sides of triangle $ABC$.
Solution. Let's find one of the segments, since the others are found similarly.

Since $\frac{A I}{I L_{1}}=\frac{b+c}{a}$, then:

![](https://cdn.mathpix.com/cropped/2024_05_21_c833ac06d3d78911cbb1g-076.jpg?height=509&width=674&top_left_y=1770&top_left_x=1022)

$$
\frac{A I}{A L_{1}}=\frac{b+c}{a+b+c}
$$

From the similarity of triangles $A E F$ and $A C B$ (Fig. 101) we have:

$$
\frac{E F}{B C}=\frac{A I}{A L_{1}} \text { or } \frac{E F}{a}=\frac{b+c}{a+b+c} \text { hence } E F=\frac{a(b+c)}{a+b+c}
$$

(Note that it is not difficult to show that $C E+B F=E F$ (Fig. 101), since $C I$ and $B I$ are segments of the angle bisectors. Then $C E+B F=\frac{a(b+c)}{a+b+c}$ and another formulation of the problem 99 is possible: find the sum of the segments $C E+B F$).