"Trapezoid In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid () in American and Canadian English but as a trapezium () in English outside North America. The parallel sides are called the ""bases"" of the trapezoid and the other two sides are called the ""legs"" or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A ""scalene trapezoid"" is a trapezoid with no sides of equal measure, in contrast to the special cases below. The term ""trapezium"" has been in use in English"
"Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two Latin words ""quadri"", a variant of four, and ""latus"", meaning ""side"". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called 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"
"Antiparallelogram In geometry, an antiparallelogram is a quadrilateral having, like a parallelogram, two opposite pairs of equal-length sides, but in which the sides of one pair cross each other. The longer of the two pairs will always be the one that crosses. Antiparallelograms are also called contraparallelograms or crossed parallelograms. An antiparallelogram is a special case of a crossed quadrilateral, which has generally unequal edges. A special form of the antiparallelogram is a crossed rectangle, in which two opposite edges are parallel. Every antiparallelogram has an axis of symmetry through its crossing point. Because of this symmetry, it has two"
"This is a special case of the ""n""-gon interior angle sum formula (""n"" − 2) × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges. Any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all interior angles are less than 180° and the two diagonals both lie inside the quadrilateral. In a concave quadrilateral, one interior angle is bigger than 180° and one of the two diagonals lies outside the quadrilateral. A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a"
"a ""puckered"" ring of four atoms. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral. A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed. Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two"
"Parallelogon A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides; a four-sided parallelogon is a parallelogram. In general a parallelogon has 180-degree rotational symmetry around its center. Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. In general they all have central inversion symmetry, order 2. Every convex"
"(orbifold notation), and include: Saccheri quadrilateral A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book ""Euclides ab omni naevo vindicatus"" (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum. The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral. For a Saccheri quadrilateral"
"Square In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ""ABCD"" would be denoted . A convex quadrilateral is a square if and only if it is any one of the following: A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides"
"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 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 Hence 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"
"Kite (geometry) In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as deltoids, but the word ""deltoid"" may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be"
"Equidiagonal quadrilateral In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types. Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12. A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of"
"Rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is a parallelogram and"
"an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia). A"
"Cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ""circumcircle"" or circumscribed circle, and the vertices are said to be ""concyclic"". The center of the circle and its radius are called the ""circumcenter"" and the ""circumradius"" respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below"
"or more as a truncation of 4 sequential vertices. Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length. Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid."
"a tangential quadrilateral ""ABCD"" is also a trapezoid with parallel sides ""AB"" and ""CD"" if and only if and ""AD"" and ""BC"" are the parallel sides of a trapezoid if and only if Tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the ""incenter"" and its radius is called the ""inradius"". Since these quadrilaterals can be drawn surrounding or circumscribing their incircles,"
"diagonals, in the form An inequality for the product of the diagonals ""p"", ""q"" in a bicentric quadrilateral is where ""a"", ""b"", ""c"", ""d"" are the sides. This was proved by Murray S. Klamkin in 1967. Bicentric quadrilateral In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called the ""inradius"" and ""circumradius"", and ""incenter"" and ""circumcenter"" respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent"
"Levi-Civita parallelogramoid In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane. It is named for its discoverer, Tullio Levi-Civita. Like a parallelogram, two opposite sides ""AA""′ and ""BB""′ of a parallelogramoid are parallel (via parallel transport side ""AB"") and the same length as each other, but the fourth side ""A""′""B""′ will not in general be parallel to or the same length as the side ""AB,"" although it will be straight (a geodesic). A parallelogram in Euclidean geometry can be constructed as"
"has 10 faces on the polar axis with 10 faces following the equator. Rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically"
"""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: The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that"
"pairs of equal angles as well as two pairs of equal sides. Together with the kites and the isosceles trapezoids, antiparallelograms form one of three basic classes of quadrilaterals with a symmetry axis. The convex hull of an antiparallelogram is an isosceles trapezoid, and every antiparallelogram may be formed from the non-parallel sides (or either pair of parallel sides in case of a rectangle) and diagonals of an isosceles trapezoid. Every antiparallelogram is a cyclic quadrilateral, meaning that its four vertices all lie on a single circle. Several nonconvex uniform polyhedra, including the tetrahemihexahedron, cubohemioctahedron, octahemioctahedron, small rhombihexahedron, small icosihemidodecahedron,"
"Varignon parallelogram (whose vertices are the midpoints of its sides) is a rectangle. A related characterization states that a convex quadrilateral is orthodiagonal if and only if the midpoints of the sides and the feet of the four maltitudes are eight concyclic points; the eight point circle. The center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the ""principal orthic quadrilateral"". If the normals to the sides of a convex quadrilateral ""ABCD"" through the diagonal intersection intersect the opposite sides in ""R"", ""S"", ""T"", ""U"", and ""K"", ""L"","
"is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles. Four lengths ""a"", ""c"", ""b"", ""d"" can constitute the consecutive sides of a non-parallelogram trapezoid with ""a"" and ""b"" parallel only when The quadrilateral is a parallelogram when formula_2, but it is an ex-tangential quadrilateral (which is not a trapezoid) when formula_3. Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: Additionally, the following properties are equivalent,"
"that is also a parallelogram). A square is a limiting case of both a kite and a rhombus. Orthodiagonal equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the ""n"" = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others. An orthodiagonal quadrilateral that is also equidiagonal is a midsquare quadrilateral because its Varignon parallelogram is a square. Its area can be expressed purely in terms of its sides. For"
"each diagonal. An automedian triangle is one whose medians are in the same proportions as its sides (though in a different order). If ""ABC"" is an automedian triangle in which vertex ""A"" stands opposite the side ""a"", ""G"" is the centroid (where the three medians of ""ABC"" intersect), and ""AL"" is one of the extended medians of ""ABC"" with ""L"" lying on the circumcircle of ""ABC"", then ""BGCL"" is a parallelogram. The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. If the quadrilateral is convex or concave (that is, not"
"any pair of conjugate diameters, or from any tangent parallelogram. A parallelepiped is a three-dimensional figure whose six faces are parallelograms. Parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a"
"Orthodiagonal quadrilateral In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other. A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals. A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral"
"of opposite parallel sides and the diagonals of a parallelogram, the Varignon parallelogram is a line segment traversed twice. Varignon's theorem Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, who published it in 1731. The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the area of the quadrilateral. If one introduces the concept of oriented areas for"
"three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology (in Greek παραλληλ-όγραμμον, a shape ""of parallel lines"") reflects the definition. A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true: Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram. All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms: A parallelogram with base ""b"" and height ""h"" can be divided into a trapezoid"
"the midpoint of the opposite side. An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include the square, the rhombus, and the kite. By Brahmagupta's theorem, in an orthodiagonal quadrilateral that is also cyclic, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side. By van Aubel's theorem, if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length. Up to three lines in three-dimensional space can be pairwise perpendicular, as"
"is a parallelogram, then the midpoints of the diagonals coincide so that the connecting line segment formula_4 has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to which is the parallelogram law. If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: Dividing by 2 yields the Euler–Pythagoras theorem: In other words, in the case of a rectangle the relation of the quadrilateral's sides and its diagonals is described by the Pythagorean theorem. Euler originally derived the theorem above as corollary from slightly"
"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°. 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 below). The four maltitudes of a convex quadrilateral are the perpendiculars to a side through the midpoint of the"
"above formulas. Whence 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: The Varignon parallelogram is a rhombus if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral. The Varignon parallelogram is a rectangle if and only if the diagonals of the quadrilateral are perpendicular, that is, if the quadrilateral is an orthodiagonal quadrilateral. If a crossing quadrilateral is formed from either pair"