The difference in the radii of two externally tangent circles is $5 \mathrm{~cm}$; the distance from the intersection point of the common tangent and the line connecting the centers to the point of tangency on the smaller circle is 2.4 times the radius of the smaller circle. What are the radii of the two circles?
Let the radius of the smaller circle be denoted by $\varrho$, and the radius of the larger circle will be $\varrho+5$. The distance between the centers $O_{1}$ and $O_{2}$ is $2 \varrho+5$. The length of the segment $s$ from the point of tangency of the smaller circle to the intersection point $M$ with the central line is $2.4 \varrho$ (see the first diagram).

![](https://cdn.mathpix.com/cropped/2024_05_02_064ab427ac35faf3c663g-1.jpg?height=376&width=737&top_left_y=262&top_left_x=694)

The hypotenuse $O_{2} M$ of the right triangle formed by the radius $\varrho$ at the point of tangency can be calculated using the Pythagorean theorem:

$$
O_{2} M=\sqrt{(2.4 \varrho)^{2}+\varrho^{2}}=\sqrt{6.76 \varrho^{2}}=2.6 \varrho
$$

The radii drawn to the points of tangency create two similar right triangles, and the ratio of the corresponding sides is the same:

$$
\varrho: 2.6 \varrho=(\varrho+5):(2 \varrho+5+2.6 \varrho)
$$

Multiplying the means and the extremes, we get:

$$
2.6 \varrho^{2}+13 \varrho=4.6 \varrho^{2}+5 \varrho
$$

After simplifying and dividing the equation by $\varrho (\neq 0)$, we obtain:

$$
\varrho=4
$$

Thus, the radius of the smaller circle is $4 \text{ cm}$, and the radius of the larger circle is $9 \text{ cm}$.

Etelka Angermayer (Mosonmagyaróvár, Kossuth g. I. o. t.)