Question

Two circle touch both the arms of an angle whose measure is ${60}^o$. Both the circles also touch each other externally. The radius of the smaller circle is $r$. The radius of the bigger circle (in term of $r$ ) is

Answer 1

$\begin{array}{c}In\Delta OAB\\ \Rightarrow \frac{R}{R+2r+PB}=\sin {30}^0\\ \Rightarrow \frac{R}{R+2r+PB}=\frac{1}{2}\\ \Rightarrow 2R=R+2r+PB\\ \Rightarrow R-2r=PB\to \left(i\right)\\ In\Delta OCB\\ \Rightarrow \frac{r}{r+PB}=\sin {30}^0\\ \Rightarrow \frac{r}{r+PB}=\frac{1}{2}\\ \Rightarrow 2r=r+PB\\ fromeq.\left(i\right)\\ \Rightarrow r=R-2r\\ \therefore R=3r\end{array}$