Problem 100. The bisectors $A L_{1}$ and $B L_{2}$ of triangle $A B C$ intersect at point I. It is known that $A I: I L_{1}=3 ; B I: I L_{2}=2$. Find the ratio of the sides in triangle $A B C$.
Solution. According to the property of the incenter $\frac{b+c}{a}=3 ; \frac{a+c}{b}=2$, or $b+c=3a ; a+c=2b$. Solving this system of equations, we find the sides of the triangle: $\frac{3}{5} c ; \frac{4}{5} c ; c$. That is, the ratio of the lengths of the sides of the triangle is: $3: 4: 5^{1}$.