2017 AMC 10B Problem 24

The vertices of an equilateral triangle lie on the hyperbola xy=1, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

\textbf{(A)}\ 48\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 108\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 169

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