\section*{Problem 5 - 091235}

The plane \(\epsilon\) of a given triangle \(\triangle A B C\) is touched at its vertices by three spheres in such a way that the spheres also touch each other pairwise from the outside.

Determine the radii of the three spheres in dependence on the side lengths of the given triangle!
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Let \(M_{A}, M_{B}, M_{C}\) and \(r_{A}, r_{B}, r_{C}\) be the centers and radii of the spheres, respectively. The quadrilateral \(A B M_{B} M_{A}\) has right angles at A and B, and the sides \(A M_{A}\) and \(B M_{B}\) are parallel. Since the spheres touch each other, \(M_{A} M_{B}\) has the length \(r_{A}+r_{B}\). The Pythagorean theorem yields the equation \(|A B|^{2}+\left(\left|A M_{A}\right|-\left|B M_{B}\right|\right)^{2}=\) \(\left|M_{A} M_{B}\right|^{2}\).

Substitution and simplification yield the equation \(c^{2}=4 r_{A} r_{B}\). Together with the equations for the other two sides, we obtain the system of equations

\[
c^{2}=4 r_{A} r_{B} \quad b^{2}=4 r_{A} r_{C} \quad a^{2}=4 r_{B} r_{C}
\]

This system has the solution \(r_{A}=\frac{b c}{2 a}, r_{B}=\frac{a c}{2 b}, r_{C}=\frac{a b}{2 c}\)

Source: anonymous