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 \(A\), \(B\), \(C\) and \(D\) is sometimes denoted as \(\square ABCD\).

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, that is

\(\angle A+\angle B+\angle C+\angle D=360^{\circ}.\)

This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180° (here, n=4).

All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.

In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.

• Irregular quadrilateral: no sides are parallel.
• Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.
• Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
• Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
• Rhombus, rhomb: all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus.
• Rectangle: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
• Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
• Oblong: longer than wide, or wider than long (i.e., a rectangle that is not a square).
• Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi. It is a type of tangential quadrilateral.

Condensed: the full section is in Wikipedia.

In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.

• A dart (or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See Kite.

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°.

• Crossed trapezoid (US) or trapezium (Commonwealth): a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a trapezoid).
• Antiparallelogram: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a parallelogram).
• Crossed rectangle: an antiparallelogram whose sides are two opposite sides and the two diagonals of a rectangle, hence having one pair of parallel opposite sides.
• Crossed square: a special case of a crossed rectangle where two of the sides intersect at right angles.

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. 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.

The area can be expressed in trigonometric terms as

\(K = \tfrac12 pq \sin \theta,\)

where the lengths of the diagonals are p and q and the angle between them is θ. In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to \(K=\tfrac{pq}{2}\) since θ is 90°.

The area can be also expressed in terms of bimedians as

\(K = mn \sin \varphi,\)

where the lengths of the bimedians are m and n and the angle between them is φ.

Bretschneider's formula expresses the area in terms of the sides and two opposite angles:

\(\begin{aligned} K &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (A + C) ]} \\ &= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd\, \cos^2 \tfrac12(A + C) } \end{aligned}\)

where the sides in sequence are a, b, c, d, where s is the semiperimeter, and A and C are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral, when A + C = 180° .

Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is

\(K = \tfrac12 ad \sin{A} + \tfrac12 bc \sin{C}.\)

\(K = \tfrac14 \left|\tan \theta\right| \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.\)

\(K=\tfrac12 \sqrt{\bigl((a^2+c^2)-2x^2\bigr)\bigl((b^2+d^2)-2x^2\bigr)} \sin{\varphi}\)

\(K=\tfrac12 ab \sin{\alpha}+\tfrac14 \sqrt{4c^2d^2-(c^2+d^2-a^2-b^2+2ab \cos{\alpha})^2} ,\)

Condensed: the full section is in Wikipedia.

The following two formulas express the area in terms of the sides a, b, c and d, the semiperimeter s, and the diagonals p, q:

\(K = \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},\)

\(K = \tfrac14 \sqrt{4p^2q^2 - \left( a^2 + c^2 - b^2 - d^2 \right)^2}.\)

The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then pq = ac + bd.

The area can also be expressed in terms of the bimedians m, n and the diagonals p, q:

\(K=\tfrac12 \sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},\)

\(K=\tfrac12 \sqrt{p^2q^2-(m^2-n^2)^2}.\)

In fact, any three of the four values m, n, p, and q suffice for determination of the area, since in any quadrilateral the four values are related by \(p^2+q^2=2(m^2+n^2).\) The corresponding expressions are:

\(K=\tfrac12 \sqrt{[(m+n)^2-p^2]\cdot[p^2-(m-n)^2]},\)

if the lengths of two bimedians and one diagonal are given, and

\(K=\tfrac14 \sqrt{[(p+q)^2-4m^2]\cdot[4m^2-(p-q)^2]},\)

if the lengths of two diagonals and one bimedian are given.

The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then

\(K = \tfrac12 |\mathbf{AC}\times\mathbf{BD}|,\)

which is half the magnitude of the cross product of vectors AC and BD. In two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x₁,y₁) and BD as (x₂,y₂), this can be rewritten as:

\(K = \tfrac12 |x_1 y_2 - x_2 y_1|.\)

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. The list applies to the most general cases, and excludes named subsets.

• Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral.
• Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).

The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines on each triangle formed by one diagonal and two sides of the quadrilateral. Thus

\(p=\sqrt{a^2+b^2-2ab\cos{B}}=\sqrt{c^2+d^2-2cd\cos{D}}\)

and

\(q=\sqrt{a^2+d^2-2ad\cos{A}}=\sqrt{b^2+c^2-2bc\cos{C}}.\)

Other, more symmetric formulas for the lengths of the diagonals, are

\(p=\sqrt{\frac{(ac+bd)(ad+bc)-2abcd(\cos{B}+\cos{D})}{ab+cd}}\)

and

\(q=\sqrt{\frac{(ab+cd)(ac+bd)-2abcd(\cos{A}+\cos{C})}{ad+bc}}.\)

In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus

\(a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2\)

where x is the distance between the midpoints of the diagonals. This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law.

The German mathematician Carl Anton Bretschneider derived in 1842 the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateral

\(p^2q^2=a^2c^2+b^2d^2-2abcd\cos{(A+C)}.\)

This relation can be considered to be a law of cosines for a quadrilateral. In a cyclic quadrilateral, where A + C = 180°, it reduces to pq = ac + bd. Since cos (A + C) ≥ −1, it also gives a proof of Ptolemy's inequality.

If X and Y are the feet of the normals from B and D to the diagonal AC = p in a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, then

\(XY=\frac{|a^2+c^2-b^2-d^2|}{2p}.\)

In a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, and where the diagonals intersect at E,

\(efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg)\)

where e = AE, f = BE, g = CE, and h = DE.

The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p, q and the four side lengths a, b, c, d of a quadrilateral are related by the Cayley-Menger determinant, as follows:

\(\det \begin{bmatrix} 0 & a^2 & p^2 & d^2 & 1 \\ a^2 & 0 & b^2 & q^2 & 1 \\ p^2 & b^2 & 0 & c^2 & 1 \\ d^2 & q^2 & c^2 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{bmatrix} = 0.\)

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (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.

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.

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:

• Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
• A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
• The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.
• The perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
• The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.

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.

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

\(m=\tfrac{1}{2}\sqrt{-a^2+b^2-c^2+d^2+p^2+q^2}\)

where p and q are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides b and d is

\(n=\tfrac{1}{2}\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.\)

Hence

\(\displaystyle p^2+q^2=2(m^2+n^2).\)

This is also a corollary to the parallelogram law applied in the Varignon parallelogram.

\(m=\tfrac{1}{2}\sqrt{2(b^2+d^2)-4x^2}\)

\(n=\tfrac{1}{2}\sqrt{2(a^2+c^2)-4x^2}.\)

• The two bimedians have equal length if and only if the two diagonals are perpendicular.
• The two bimedians are perpendicular if and only if the two diagonals have equal length.

Condensed: the full section is in Wikipedia.