264. Given an equilateral triangle. Determine the radius of the inscribed and circumscribed circles.
264. Let the side of the triangle be $a$ (Fig. 63). Then it is clear that $r=\frac{R}{2}$, and also, $R^{2}=r+\frac{a^{2}}{4}=\frac{R^{2}}{4}+\frac{a^{2}}{4}$, from which

\[
\begin{gathered}
3 R^{2}=a^{2} ; R^{2}=\frac{a^{2}}{3} ; R \frac{a}{\sqrt{3}}=\frac{a \sqrt{3}}{3} . \\
r=\frac{R}{2}=\frac{a \sqrt{3}}{6} .
\end{gathered}
\]

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-139.jpg?height=405&width=417&top_left_y=1391&top_left_x=480)

Fig. 63.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-139.jpg?height=509&width=528&top_left_y=1276&top_left_x=1047)

Fig. 64.