2003 AMC 12A Problem 16

A point P is chosen at random in the interior of equilateral triangle ABC. What is the probability that \triangle ABP has a greater area than each of \triangle ACP and \triangle BCP?

\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{2}{3}

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