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? 


In a equilateral triangle the ratio $\frac{A_{\text{incircle}}}{A_{\text{triangle}}}$ equals $\frac{\pi}{3\sqrt{3}}$. Step 0: we draw the incircle of the original triangle.
Step 1: we draw three incircles for equilateral triangles with side length $\frac{1}{3}$. Step $n$: we draw three incircles for equilateral triangles with side length $\frac{1}{3^n}$. The area occupied by the circles is so
$$ \frac{\pi}{3\sqrt{3}}\cdot \frac{\sqrt{3}}{4}\cdot 1^2 + \sum_{n\geq 1}\frac{\pi}{3\sqrt{3}}\cdot \frac{\sqrt{3}}{4}\cdot 3\left(\frac{1}{3^n}\right)^2 $$
which simplifies to
$$ \frac{\pi}{12}+\frac{\pi}{12}\sum_{n\geq 1}\frac{3}{9^n}=\frac{\pi}{12}+\frac{\pi}{12}\cdot\frac{3}{8}=\frac{11\pi}{96}. $$
