Ex. 129. In an integer-sided triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer.
Ex. 129. Answer: 12. Solution. Let the third side be denoted by $a$, and the angle subtending it by $\alpha$. Then $\sin \alpha=\frac{10+10+a}{10 \cdot 10} \cdot r$. At the same time, $1 \leq a \leq 19 \Rightarrow r \leq 4$. If $\sin \alpha=1$, then $x=5, r=4$, but a triangle with such data does not exist. According to the result of Ex. 124, $\sin \alpha=\frac{m}{n}$, where $n$ is either 5 or 25. In the first case, either $\sin \alpha=\frac{3}{5}$ or $\sin \alpha=\frac{4}{5}$. In both cases, a triangle with such data does not exist. If $n=25$, then $\sin \alpha=\frac{7}{25}$ or $\frac{24}{25}$. After verification, we get $a=12, r=3$.