The sides of the triangle are equal to $a, b, c$. Three spheres touch each other pairwise and the plane of the triangle at its vertices. Find the radii of the spheres.
Let spheres of radii $x, y$, and $z$ with centers $A_{1}, B_{1}$, and $C_{1}$ touch the plane of triangle $ABC$ with sides $AB=c$, $AC=b$, and $BC=a$ at vertices $A, B$, and $C$ respectively (Fig. 1). Draw a plane through the parallel lines $C C_{1}$ and $B B_{1}$. We obtain two tangent circles of radii $y$ and $z$ and a line tangent to these circles at points $B$ and $C$ (Fig. 2). Drop a perpendicular $B_{1} F$ from point $B_{1}$ to line $C C_{1}$. Then

$C_{1} F=\left|C C_{1}-C F\right|=\left|C C_{1}-B B_{1}\right|=|z-y|$

$a^{2}=B C^{2}=B_{1} F^{2}=B_{1} C_{1}{ }^{2}-C_{1} F^{2}=(z+y)^{2}-(z-y)^{2}=4 y z$.

Similarly, $b^{2}=4 x z$ and $c^{2}=4 x y$. Thus, we have the system of equations for $x, y$, and $z$:

$$
\left\{\begin{array}{c}
4 \\
x z=\begin{array}{c}
b^{2} \\
4 \\
x y= \\
c^{2} \\
4
\end{array}
\end{array}\right.
$$

Multiplying the second and third equations term by term and dividing the result by $b^{2} c^{2}$, we get $x^{2}=$

$4 a^{2}$. Therefore,

$x=\begin{aligned} & b c \\ & 2 a\end{aligned}$. Similarly, $y=\begin{aligned} & a c \\ & 2 b\end{aligned}, z=\begin{aligned} & a b \\ & 2 c\end{aligned}$.