7. A circle is drawn through two vertices of an equilateral triangle $A B C$ with an area of $21 \sqrt{3} \mathrm{~cm}^{2}$, for which two sides of the triangle are tangents. Find the radius of this circle.
Solution: $S_{\triangle A B C}=21 \sqrt{3}, R=$ ? ,

$S_{\triangle A B C}=\frac{a^{2} \sqrt{3}}{4} \Rightarrow \frac{a^{2} \sqrt{3}}{4}=21 \sqrt{3}, a^{2}=84, a=2 \sqrt{21}$.

$\angle C A O=90^{\circ}-\angle B A C=30^{\circ}, \frac{a}{2}=R \cos 30^{\circ}, \sqrt{21}=R \frac{\sqrt{3}}{2}, 2 \sqrt{7}=R$.

Answer: $R=2 \sqrt{7}$.