2018 Fermat Problem 16

Points P and Q are two distinct points in the xy-plane. In how many different places in the xy-plane can a third point, R, be placed so that PQ = QR = PR?

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

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.