2017 AMC 10A Problem 22

Sides \overline{AB} and \overline{AC} of equilateral triangle ABC are tangent to a circle at points B and C respectively. What fraction of the area of \triangle ABC lies outside the circle?

\textbf{(A)}\ \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad\textbf{(B)}\ \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad\textbf{(E)}\ \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.