A1. In an equilateral triangle with a side length of 1, three equal circles are inscribed (see figure). What is the radius of the circles?
(A) $\frac{1}{3}$
(B) $\frac{1}{4}$
(C) $\frac{\sqrt{3}}{4}$
(D) $\frac{\sqrt{3}+1}{4}$
(E) $\frac{\sqrt{3}-1}{4}$

![](https://cdn.mathpix.com/cropped/2024_06_07_4050f14f84cf4f327f67g-06.jpg?height=195&width=229&top_left_y=762&top_left_x=1690)
A1. If we denote the radii of the circles by $\mathbf{z} r$, then $r \sqrt{3}+2 r+r \sqrt{3}=1$. It follows that $r=\frac{1}{2(1+\sqrt{3})}=\frac{\sqrt{3}-1}{4}$.