8

Two touching circles with centers $O_{1}$ and $O_{2}$ touch internally a circle of radius $R$ with center $O$. Find the perimeter of triangle $O O_{1} O_{2}$.
Let $A_{1}, A_{2}$ and $B$ be the points of tangency of the circles with centers $O$ and $O_{1}, O$ and $O_{2}, O_{1}$ and $O_{2}$, respectively. Then $O_{1} O_{2}=O_{1} B+B O_{2}=$ $O_{1} A_{1}+O_{2} A_{2}$. Therefore, $O O_{1}+OO_{2}+O_{1} O_{2}=(OO_{1}+O_{1} A_{1})+(OO_{2}+O_{2} A_{2})=O A_{1}+O A_{2}=2 R$.