10.048. Two circles are inscribed in an acute angle of $60^{\circ}$, touching each other externally. The radius of the smaller circle is $r$. Find the radius of the larger circle.
## Solution.

By the condition $\angle B A C=60^{\circ}, O_{1} C=r$ (Fig. 10.48). In $\triangle A C O_{1} \angle A C O_{1}=90^{\circ}$, $\angle O_{1} A C=30^{\circ} ; A O_{1}=\frac{O_{1} C}{\sin 30^{\circ}}=2 O_{1} C=2 r$. In $\triangle O_{2} E A \quad \angle O_{2} E A=90^{\circ}$; $\Delta O_{2} E A \sim \Delta O_{1} C A \Rightarrow \frac{O_{2} E}{O_{1} C}=\frac{3 r+R}{2 r} ; \frac{R}{r}=\frac{3 r+R}{2 r} ; R=3 r$.

Answer: $3 r$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-579.jpg?height=486&width=392&top_left_y=75&top_left_x=172)

Fig. 10.49

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-579.jpg?height=388&width=402&top_left_y=188&top_left_x=799)

Fig. 10.50