A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$? $\textbf{(A)} - \!\frac {1}{3} \qquad$ $ \textbf{(B)} - \!\frac {1}{6} \qquad$ $ \textbf{(C)}\ \frac {1}{6} \qquad$ $ \textbf{(D)}\ \frac {1}{3} \qquad$ $ \textbf{(E)}\ \frac {1}{2}$