Equations of a straight line

If the slope of line DE  =  the slope of line BC, then DE  is parallel to  BC.

D  is the midpoint of AB, so    D   =   $(\frac{4+2}{2},\frac{6-2}{2})$   =   (3, 2)

E  is the midpoint of  AC, so   E   =   $(\frac{4-2}{2},\frac{6-4}{2})$   =   (1, 1)

slope of  DE  =  $\frac{y_2-y_1}{x_2-x_1}\,=\,\frac{2-1}{3-1}\,=\,\frac{1}{2}$

slope of  BC  =  $\frac{y_2-y_1}{x_2-x_1}\,=\,\frac{-4+2}{-2-2}\,=\,\frac{-2}{-4}\,=\,\frac{1}{2}$

Since the slope of  DE  and the slope of  BC  both equal  $\frac12$ ,  DE  is parallel to  BC.

Here's a graph to check:  https://www.desmos.com/calculator/5k2mf7pbjj