Quadrilateral - Remarkable Points and Lines in A Convex Quadrilateral

Remarkable Points and Lines in A Convex Quadrilateral

The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just centroid (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.

The "vertex centroid" is the intersection of the two bimedians. As with any polygon, the x and y coordinates of the vertex centroid are the arithmetic means of the x and y coordinates of the vertices.

The "area centroid" of quadrilateral ABCD can be constructed in the following way. Let G_a, G_b, G_c, G_d be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines G_aG_c and G_bG_d.

In a general convex quadrilateral ABCD, there are no natural analogies to the circumcenter and orthocenter of a triangle. But two such points can be constructed in the following way. Let O_a, O_b, O_c, O_d be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by H_a, H_b, H_c, H_d the orthocenters in the same triangles. Then the intersection of the lines O_aO_c and O_bO_d is called the quasicircumcenter; and the intersection of the lines H_aH_c and H_bH_d is called the quasiorthocenter of the convex quadrilateral. These points can be used to define an Euler line of a quadrilateral. In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order, and HG = 2GO.

There can also be defined a quasinine-point center E as the intersection of the lines E_aE_c and E_bE_d, where E_a, E_b, E_c, E_d are the nine-point centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH.

Another remarkable line in a convex quadrilateral is the Newton line.