## Task A-2.2.

Two circles with radii 1 and 3 touch each other externally at point $A$, and their external common tangent touches them at points $B$ and $C$. Determine the sum of the squares of the lengths of the sides of triangle $A B C$.
## Solution.

Let $k_{1}$ and $k_{2}$ be the smaller and larger circles, respectively. Let $S_{1}$ and $S_{2}$ be the centers of circles $k_{1}$ and $k_{2}$, and let $T$ be the intersection of line $B C$ and their common tangent through $A$.

Since points $B$ and $C$ lie on circles $k_{1}$ and $k_{2}$, respectively, we have $\left|S_{1} B\right|=1$ and $\left|S_{2} C\right|=3$. Point $A$ lies on segment $\overline{S_{1} S_{2}}$, so $\left|S_{1} S_{2}\right|=\left|S_{1} A\right|+\left|A S_{2}\right|=1+3=4$.

Point $T$ lies on the tangents of circle $k_{1}$, so it is equidistant from the points of tangency, i.e., $|T A|=|T B|$. Similarly, it lies on the tangents of circle $k_{2}$, so $|T A|=|T C|$.

We conclude that point $T$ is the center of the circumcircle of triangle $A B C$. Additionally, point $T$ lies on segment $\overline{B C}$, so we conclude that triangle $A B C$ is a right triangle with a right angle at vertex $A$.

By the Pythagorean theorem, we have

$$
|A B|^{2}+|B C|^{2}+|C A|^{2}=2|B C|^{2}
$$

so it remains to determine the length of segment $\overline{B C}$.

Let $P$ be the orthogonal projection of $S_{1}$ onto $S_{2} C$. Segments $\overline{S_{1} B}$ and $\overline{S_{2} C}$ are perpendicular to line $B C$ because $B$ and $C$ are points of tangency of the tangent and the circles. Therefore, quadrilateral $S_{1} B C P$ has three right angles, making it a rectangle. Specifically, we have

$$
\begin{gathered}
\left|S_{1} P\right|=|B C| \\
|P C|=\left|S_{1} B\right|=1 \\
\left|S_{2} P\right|=\left|S_{2} C\right|-|P C|=3-1=2
\end{gathered}
$$

Finally, from the right triangle $S_{1} P S_{2}$, we have

$$
|B C|^{2}=\left|S_{1} P\right|^{2}=\left|S_{1} S_{2}\right|^{2}-\left|S_{2} P\right|^{2}=4^{2}-2^{2}=12
$$

from which it follows that

$$
|A B|^{2}+|B C|^{2}+|C A|^{2}=2|B C|^{2}=24
$$