183. To determine the side of a regular dodecagon inscribed in a circle.

Al-Karhi gives a verbal expression, according to which:

$$
a_{12}^{2}=\left[\frac{d}{2}-\sqrt{\left(\frac{d}{2}\right)^{2}-\left(\frac{d}{4}\right)^{2}}\right]^{2}+\left(\frac{d}{4}\right)^{2}
$$

Verify the validity of this expression.

Note. Problems 164-183 are taken from the treatises "Kafi fil-Hisab" and "Al-Fakri".

## Problem from the arithmetic treatise of Thabit ibn-Qurra.
183. Examining the drawing, the reader will easily find that

\[
\begin{gathered}
a_{12}^{2}=\left[\frac{d}{2}-\sqrt{\frac{3}{16} a^{2}}\right]^{2}+\left(\frac{d}{4}\right)^{2}=\frac{d^{2}}{4}+{ }_{16}^{3} d^{2}-d_{4}^{2} \sqrt{3}+ \\
+\frac{d^{2}}{16}=d^{2}\left(\frac{1}{2}-\frac{\sqrt{3}}{4}\right)=\frac{d^{2}}{4}(2-\sqrt{3})=r^{2}(2-\sqrt{3}) \\
a_{12}=r \sqrt{2-\sqrt{3}}
\end{gathered}
\]

Thabit ibn Qurra (836-901) is known as a translator of works by Euclid, Archimedes, Apollonius, and Ptolemy into Arabic. One of his original works that has come down to us is dedicated to the study of various properties of numbers.