VARIOUS CENTROIDS OF POLYGONS AND SOME CHARACTERIZATIONS OF RHOMBI Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.4134/ckms.c160023
· OA: W2609819258
For a polygon P, we consider the centroid <TEX>$G_0$</TEX> of the vertices of P, the centroid <TEX>$G_1$</TEX> of the edges of P and the centroid <TEX>$G_2$</TEX> of the interior of P. When P is a triangle, (1) we always have <TEX>$G_0=G_2$</TEX> and (2) P satisfies <TEX>$G_1=G_2$</TEX> if and only if it is equilateral. For a quadrangle P, one of <TEX>$G_0=G_2$</TEX> and <TEX>$G_0=G_1$</TEX> implies that P is a parallelogram. In this paper, we investigate the relationships between centroids of quadrangles. As a result, we establish some characterizations for rhombi and show that among convex quadrangles whose two diagonals are perpendicular to each other, rhombi and kites are the only ones satisfying <TEX>$G_1=G_2$</TEX>. Furthermore, we completely classify such quadrangles.