<s>[INST] Finding the area of circles in triangle In the figure there are infinitely many circles approaching the
vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of
length $1$, how can i find the total area occupied by the circles? 

 [/INST] Denote the circumradius by $R$ and the radii of the circles by $r_1,r_2,r_3,...$. 

We will find $R$ and $r_1$:
$$r_1=\frac{2S}{a+b+c}=\frac{2\cdot\frac{\sqrt{3}}{4}}{3}=\frac{1}{2\sqrt{3}} \\
R=\frac{abc}{4S}=\frac{1}{4\cdot \frac{\sqrt{3}}{4}}=\frac{1}{\sqrt{3}}.$$
From the similarity of the triangles:
$$\begin{align}\frac{r_2}{r_1}=\frac{R-r_1-r_2}{R} \Rightarrow r_2=&\frac{1}{2\cdot 3\sqrt{3}}, \\
\frac{r_3}{r_1}=\frac{R-r_1-2r_2-r_3}{R} \Rightarrow r_3=&\frac{1}{2\cdot 3^2\sqrt{3}}, \\ 
\frac{r_4}{r_1}=\frac{R-r_1-2r_2-2r_3-r_4}{R} \Rightarrow r_4=&\frac{1}{2\cdot 3^3\sqrt{3}}, \\
\vdots \\
r_n=&\frac{1}{2\cdot 3^{n-1}\sqrt{3}}
\end{align}$$ 
Hence:
$$\begin{align}S=&\pi r_1^2+3\cdot \pi \cdot \left(r_2^2+r_3^2+r_4^2+\cdots\right)=\\
&\frac{\pi}{12}+3\cdot \pi\cdot \left(\frac{1}{4\cdot 3^3}+\frac{1}{4\cdot 3^5}+\frac{1}{4\cdot 3^7}+\cdots\right)=\\
&\frac{\pi}{12}+\frac{3\pi}{4}\cdot \frac{\frac{1}{3^3}}{1-\frac{1}{3^2}}=\\
&\frac{\pi}{12}+\frac{\pi}{32}=\\
&\frac{11\pi}{96}.\end{align}$$
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