Circles $k_{1}, k_{2}, k_{3}$, and $k_{4}$ are given with centers $S_{1}, S_{2}, S_{3}$, and $S_{4}$, respectively. Circle $k_{1}$ and circle $k_{3}$ touch all other circles externally, the radius of circle $k_{1}$ is $5 \mathrm{~cm}$, the distance between centers $S_{2}$ and $S_{4}$ is $24 \mathrm{~cm}$, and the quadrilateral $S_{1} S_{2} S_{3} S_{4}$ is a rhombus.

Determine the radii of circles $k_{2}, k_{3}$, and $k_{4}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_7f4ac78ad31cb883e531g-3.jpg?height=465&width=774&top_left_y=704&top_left_x=641)

Note: The image is illustrative only.

(E. Semerádová)
In the figure, the only possible arrangement of circles that satisfies all the conditions in the problem is shown.

Since the circles touch each other, the lengths of the sides of the quadrilateral \( S_{1} S_{2} S_{3} S_{4} \) are equal to the sums of the radii of the corresponding circles. Since this quadrilateral is a rhombus, all its sides are equal. Thus,

\[
s = r_{1} + r_{2} = r_{2} + r_{3} = r_{3} + r_{4} = r_{4} + r_{1},
\]

where \( r_{1}, r_{2}, r_{3}, r_{4} \) are the radii of the corresponding circles and \( s \) is the length of the side of the rhombus. From this, it follows that opposite circles are congruent, i.e., \( r_{1} = r_{3} \) and \( r_{2} = r_{4} \). The diagonals of the rhombus are perpendicular to each other and bisect each other. Therefore, the triangles formed by the sides and diagonals of the rhombus are right-angled and congruent.

![](https://cdn.mathpix.com/cropped/2024_04_17_7f4ac78ad31cb883e531g-3.jpg?height=472&width=876&top_left_y=1957&top_left_x=590)

The legs of the right-angled triangles have lengths \( r_{1} = r_{3} = 5 \, \text{cm} \) and \( \frac{1}{2} \left| S_{2} S_{4} \right| = 12 \, \text{cm} \). The length of the hypotenuse, i.e., the length of the side of the rhombus, is given by the Pythagorean theorem as

\[
s = \sqrt{12^2 + 5^2} = \sqrt{169} = 13 \, \text{cm}.
\]

From this, it follows that

\[
r_{2} = r_{4} = 13 - 5 = 8 \, \text{cm}.
\]

The radii of circles \( k_{1} \) and \( k_{3} \) are \( 5 \, \text{cm} \), and the radii of circles \( k_{2} \) and \( k_{4} \) are \( 8 \, \text{cm} \).

Scoring suggestion. 3 points for determining and justifying that \( r_{1} = r_{3} \) and \( r_{2} = r_{4} \); 2 points for determining the side of the rhombus; 1 point for determining the radii \( r_{2} \) and \( r_{4} \).