8. In triangle $A B C$, the lengths of the sides are known: $|A B|=12,|B C|=13,|C A|=15$. On side $A C$, a point $M$ is taken such that the radii of the circles inscribed in triangles $A B M$ and $B C M$ are equal. Find the ratio $|A M|:|M C|$.
8. Let $|A M|:|M C|=k$. The condition of equality of the radii of the circles inscribed in triangles $A B M$ and $B C M$ means that their areas are in the same ratio as their perimeters. From this, since the ratio of the areas is $k$, we get ; $B M \left\lvert\,=\frac{13 k-12}{1-k}\right.$. From this equality, in particular, it follows that $\frac{12}{13}<k<1$. Writing the cosine theorems for triangles $A B M$ and $B C M$ (with respect to angles $B M A$ and $B M C$) and eliminating the cosines of the angles from these equations, we obtain a quadratic equation for $k$ with roots $\frac{2}{3}$ and $\frac{22}{23}$. Considering the constraints for $k$, we get the answer: $k={ }_{23}^{22}$.