Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .[1]
This article is about four-sided mathematical shapes. For other uses, see Quadrilateral (disambiguation).
Quadrilateral
Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is[1]
This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180°.[2]
All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.[3]
A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°.[10]
Special line segments[edit]
The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices.
The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides.[12] They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below).
The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.[13]
Diagonals[edit]
Properties of the diagonals in quadrilaterals[edit]
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length.[26] The list applies to the most general cases, and excludes named subsets.
Angle bisectors[edit]
The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral[24]: p.127 (that is, the four intersection points of adjacent angle bisectors are concyclic) or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.
In quadrilateral ABCD, if the angle bisectors of A and C meet on diagonal BD, then the angle bisectors of B and D meet on diagonal AC.[31]
The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. The intersection of the bimedians is the centroid of the vertices of the quadrilateral.[14]
The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a parallelogram called the Varignon parallelogram. It has the following properties:
The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are concurrent and are all bisected by their point of intersection.[24]: p.125
In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is
where p and q are the length of the diagonals.[33] The length of the bimedian that connects the midpoints of the sides b and d is
Hence[24]: p.126
This is also a corollary to the parallelogram law applied in the Varignon parallelogram.
The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence[23]
and
Note that the two opposite sides in these formulas are not the two that the bimedian connects.
In a convex quadrilateral, there is the following dual connection between the bimedians and the diagonals:[29]
Remarkable points and lines in a convex quadrilateral[edit]
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.[45]
The "vertex centroid" is the intersection of the two bimedians.[46] 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.[47]
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 Oa, Ob, Oc, Od be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by Ha, Hb, Hc, Hd the orthocenters in the same triangles. Then the intersection of the lines OaOc and ObOd is called the quasicircumcenter, and the intersection of the lines HaHc and HbHd is called the quasiorthocenter of the convex quadrilateral.[47] 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.[47]
There can also be defined a quasinine-point center E as the intersection of the lines EaEc and EbEd, where Ea, Eb, Ec, Ed are the nine-point centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH.[47]
Another remarkable line in a convex non-parallelogram quadrilateral is the Newton line, which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the Newton's one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.[48]
For any quadrilateral ABCD with points P and Q the intersections of AD and BC and AB and CD, respectively, the circles (PAB), (PCD), (QAD), and (QBC) pass through a common point M, called a Miquel point.[49]
For a convex quadrilateral ABCD in which E is the point of intersection of the diagonals and F is the point of intersection of the extensions of sides BC and AD, let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. Then there holds: the straight lines NK and ML intersect at point P that is located on the side AB; the straight lines NL and KM intersect at point Q that is located on the side CD.
Points P and Q are called "Pascal points" formed by circle ω on sides AB and CD.
[50]
[51]
[52]