Something vecty

The median vector $\vec{M_{A}}$ originating from $A$ is equal to $\dfrac{1}{2}(\vec{AB} + \vec{AC})$ .

Similarly, the median vectors originating from $B$ and $C$ respectively are

$\vec{M_{B}} = \dfrac{1}{2}(\vec{BA} + \vec{BC})$ and $\vec{M_{C}} = \dfrac{1}{2}(\vec{CA} + \vec{CB})$ .

We then have that $\vec{M_{A}} + \vec{M_{B}} + \vec{M_{C}} = \dfrac{1}{2}(\vec{AB} + \vec{AC} + \vec{BA} + \vec{BC} + \vec{CA} + \vec{CB}) =$

$\dfrac{1}{2}((\vec{AB} + \vec{BA}) + (\vec{AC} + \vec{CA}) + (\vec{BC} + \vec{CB})) = \boxed{0}$ ,

since in general $\vec{XY} = -\vec{YX}$ for any distinct points $X,Y$ .