For a quadrilateral $P$, we consider the centroid $G_0$ of the vertices of $P$, the perimeter centroid $G_1$ of the edges of $P$ and the centroid $G_2$ of the interior of $P$, respectively. It is well known that $P$ satisfies $G_0=G_1$ or $G_0=G_2$ if and only if it is a parallelogram. In this paper, we investigate various quadrilaterals satisfying $G_1=G_2$. As a result, we establish some characterization theorems. One of them asserts the existence of convex quadrilaterals satisfying $G_1=G_2$ without symmetry.