let O be origin.let ABC be a triangle and A"B"C" be another triangle.

 

LHS:-

AA"+BB"+CC"=OA"-OA+OB"-OB+OC"-OC

                      =(OA"+OB"+OC")-(OA+OB+OC)----------------EQ1

if OG and OG" are  centroids then as puneet sir has said we get

                        =3OG"-3OG

                        =3(OG"-OG)

                        =3GG"--------------------RHS

yeah it is straight forward.

 

cheers!