10.051. Three circles of different radii touch each other pairwise. The segments connecting their centers form a right triangle. Find the radius of the smallest circle, if the radii of the largest and medium circles are 6 and 4 cm.
Solution.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-580.jpg?height=594&width=402&top_left_y=95&top_left_x=95)

Fig. 10.51
Fig. 10.52
![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-580.jpg?height=686&width=640&top_left_y=94&top_left_x=592)

Let $r$ be the radius of the smaller circle. Then $O_{1} O_{2}=r+4$, $O_{1} O_{3}=r+6$ (Fig. 10.51). Since $O_{2} O_{3}^{2}=O_{1} O_{2}^{2}+O_{1} O_{3}^{2}$, we have $10^{2}=(r+4)^{2}+(r+6)^{2}, r^{2}+10 r-24=0$, from which $r=2$ cm (the root $r=-12$ does not satisfy the condition).

Answer: $2 \text{ cm}$.