if and only if the diagonals are equal. This follows directly from the quadrilateral identity Sides The sides a, b, c, and d of any quadrilateral satisfy and Maximum and minimum properties Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter. The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral. Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area. This is a direct consequence of the fact that the area of a convex quadrilateral satisfies where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°. If P is an interior point in a convex quadrilateral ABCD, then From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of 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 Ga, Gb, Gc, Gd be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines GaGc and GbGd. In a general convex quadrilateral ABCD, there are no natural analogies to the circumcenter and orthocenter of a