Suppose that the quadrilateral ABCD is a square, and the centers of gravity E, F, G and H are at two-thirds of the bisectors of triangles OAB, OBC, OCD and ODA, respectively.
Therefore, EG=EO+OG, FH=FO+OH, EG=FH=2/3AB.
S 1= 1/2*EG*FH，
S2=AB*AB，
(s 1)/(S2)=( 1/2 * EG * FH)/(AB * AB)
=[( 1/2)*(2/3AB)*(2/3AB)]/(AB * AB)
=( 1/2)/(4/9)
=2/9
Write it in the ordinary way. I feel dizzy myself. I'd better not write!