A triangle's excircle radii are 3, 10, and 15, respectively. What is the perimeter of the triangle?
Solution. Let the sides of the triangle be denoted by $a, b, c$, its perimeter by $2s$, its area by $t$, the radii of the excircles by $r_{a}, r_{b}, r_{c}$, and the radius of the incircle by $r$. The following relationships are known (their proofs can be found, for example, in the problems 1470-72 of the Collection of Geometric Problems I):

$$
\begin{gathered}
s-a=\frac{t}{r_{a}}, \quad s-b=\frac{t}{r_{b}}, \quad s-c=\frac{t}{r_{c}} \\
s=\frac{t}{r}
\end{gathered}
$$

$$
t \doteq \sqrt{s(s-a)(s-b)(s-c)}
$$

By adding the equations in (1), we get

$$
s=3s-a-b-c=\frac{t}{r_{a}}+\frac{1}{r_{b}}+\frac{t}{r_{c}}
$$

Thus, using (2) and simplifying by $t$:

$$
\frac{1}{r}=\frac{1}{r_{a}}+\frac{1}{r_{b}}+\frac{1}{r_{c}}
$$

Substituting the given radii, we find $r=2$. Now, multiply the equations (1) and (2) and apply (3):

$$
t^{2}=s(s-a)(s-b)(s-c)=\frac{t^{4}}{r \cdot r_{a} \cdot r_{b} \cdot r_{c}}
$$

From this, $t=\sqrt{r \cdot r_{a} \cdot r_{b} \cdot r_{c}}=30$; substituting this into (2) gives $s=15$. Therefore, the perimeter of the triangle is 30 units. It is easily verifiable that the triangle in the problem exists; it is a right triangle with legs of 5 and 12 units, and a hypotenuse of 13 units.