For a polygon P, we consider the centroid

of the vertices of P, the centroid

of the edges of P and the centroid

of the interior of P, respectively. When P is a triangle, the centroid

always coincides with the centroid

. For the centroid

of a triangle, it was proved that the centroid

of a triangle coincides with the centroid

of the triangle if and only if the triangle is equilateral. In this paper, we study the relationships between the centroids

,

and

of a quadrangle P. As a result, we show that parallelograms are the only quadrangles which satisfy either

or

. Furthermore, we establish a characterization theorem for convex quadrangles satisfying

, and give some examples (convex or concave) which are not parallelograms but satisfy

.