5. Two circles $k_{1}$, $k_{2}$ with radius $\frac{1}{2} r$ are inscribed in a circle $k$ with radius $r$, and they touch each other. Circle $l$ touches circles $k_{1}$, $k_{2}$ externally and has an internal touch with circle $k$. Circle $m$ has an external touch with circles $k_{2}$ and $l$ and an internal touch with circle $k$. Calculate the radii of circles $l$ and $m$.
SOLUTION. Let $S, A, B, C, D$ be the centers of circles $k, k_{1}, k_{2}, l, m$ and $x, y$ the radii of circles $l$ and $m$. Point $C$ lies on the line passing through point $S$ and perpendicular to $A B$ (Fig. 4). From the right triangle $B C S$, by the Pythagorean theorem, we have

$$
\left(\frac{r}{2}+x\right)^{2}=\left(\frac{r}{2}\right)^{2}+(r-x)^{2}
$$

and from this, $x=\frac{1}{3} r$. Let $P, Q$ be the feet of the perpendiculars from point $D$ to the lines $A B$ and $S C$, and $u=|S P|$, $v=|S Q|$. If $u \neq \frac{1}{2} r$, then $B P D$ is a right triangle and by the Pythagorean theorem,

$$
\left(\frac{r}{2}+y\right)^{2}=v^{2}+\left(u-\frac{r}{2}\right)^{2}
$$

This equation also holds in the case $u=\frac{1}{2} r$.

![](https://cdn.mathpix.com/cropped/2024_04_17_00132d3cc3e29ab78bbdg-6.jpg?height=751&width=782&top_left_y=318&top_left_x=637)

Fig. 4

Similarly, from the right triangle $Q C D$ (if $Q \neq C$) or by comparing opposite sides of the rectangle (if $Q=C$), we get

$$
\left(\frac{r}{3}+y\right)^{2}=u^{2}+\left(v-\frac{2 r}{3}\right)^{2}
$$

Additionally, from the right triangle $S P D$, we have

$$
(r-y)^{2}=u^{2}+v^{2} \text {. }
$$

By subtracting equations (3) and (2), we get $\frac{4}{3} r^{2}-\frac{8}{3} r y=\frac{4}{3} v r$, thus $v=r-2 y$. Similarly, by subtracting equations (3) and (1), we get $r^{2}-3 r y=u r$ and thus $u=r-3 y$. Substituting into (3) and simplifying, we get

$$
\begin{gathered}
(r-y)^{2}=(r-3 y)^{2}+(r-2 y)^{2} \\
r^{2}-8 r y+12 y^{2}=0, \\
(r-6 y)(r-2 y)=0 .
\end{gathered}
$$

From this, it follows that $y=\frac{1}{2} r$ or $y=\frac{1}{6} r$. The circle $k_{1}$ has a radius of $\frac{1}{2} r$, and the circle $m$ has a radius of $\frac{1}{6} r$, as shown in Fig. 4. Each of these two circles touches circles $k, k_{2}$, and $l$ in the required manner.

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