G8.3 If the area of the circle $E F G$ is $\frac{(c+4 \sqrt{3}) \pi}{3}$, find the value of $c$
Let the centre be $O$, the equilateral triangle formed by the lines joining the centres be $P Q R$, the radius be $r$.
$$
\begin{array}{l}
r=O E=O P+P E=1 \sec 30^{\circ}+1=\frac{2}{\sqrt{3}}+1=\frac{2+\sqrt{3}}{\sqrt{3}} \\
\text { Area of circle }=\pi \cdot \frac{(2+\sqrt{3})^{2}}{3}=\frac{(7+4 \sqrt{3}) \pi}{3} \\
c=7
\end{array}
$$