1986 AHSME Problem 26

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the x and y axes and so that the medians to the midpoints of the legs lie on the lines y = 3x + 1 and y = mx + 2. The number of different constants m for which such a triangle exists is

\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ \text{more than 3}

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