AMC 10 2017b 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? $\textbf{(A)}\ 48 \textbf{(B)}\ 60 \textbf{(C)}\ 108 \textbf{(D)}\ 120 \textbf{(E)}\ 169$