KöMaL 2014 B4608

The perpendicular projections of the centroid $S$ of a triangle ABC on sides $BC$, $AC$ and $AB$ are $A_1$, $B_1$ and $C_1$, respectively. Prove that (with conventional notations) $a^2 \overrightarrow{SA_1} +b^2 \overrightarrow{SB_1}+c^2 \overrightarrow{SC_1} =\mathbf{0}$.