The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...
A quadrilateral, sometimes also known as a tetragon or quadrangle (Johnson 1929, p. 61) is a four-sided polygon. If not explicitly stated, all four polygon vertices are generally taken to lie in a plane. (If the points do not lie in a plane, the quadrilateral is called a skew quadrilateral.) There are three topological types of quadrilaterals (Wenninger 1983, p. 50): convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilaterals (or butterflies, or bow-ties; right figure).A quadrilateral with two sides parallel is called a trapezoid, whereas a quadrilateral with opposite pairs of sides parallel is called a parallelogram.For a planar convex quadrilateral (left figure above), let the lengths of the sides be , , , and , the semiperimeter , and the polygon diagonals and . The polygon diagonals are perpendicular iff .An equation for the sum of the squares of side lengths is(1)where is the..
A golden rhombus is a rhombus whose diagonals are in the ratio , where is the golden ratio.The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.The half-angle is given by(1)(2)(3)(4)(OEIS A195693).Labeling the smaller interior angle as and the larger as , then(5)and(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(OEIS A105199 and A137218).The diagonal lengths of a golden rhombus with edge length are given by(18)(19)(20)(21)(22)(23)(24)(25)(OEIS A121570 and A179290),the inradius by(26)and the area by(27)
For a cyclic quadrilateral, the sum of theproducts of the two pairs of opposite sides equals the product of the diagonals(1)(Kimberling 1998, p. 223).This fact can be used to derive the trigonometryaddition formulas.Furthermore, the special case of the quadrilateral being a rectangle gives the Pythagorean theorem. In particular, let , , , , , and , so the general result is written(2)For a rectangle, , , and , so the theorem gives(3)
The figure formed when the midpoints of the sides of a convex quadrilateral are joined in order is a parallelogram. Equivalently, the bimedians bisect each other.The area of the Varignon parallelogram of a convex quadrilateral is half that of the quadrilateral, and the perimeter is equal to the sum of the diagonals of the original quadrilateral.
For a quadrilateral which is not cyclic,Ptolemy's theorem becomes an inequality:The Ptolemy inequality is still valid when is a triangular pyramid (Boomstra 1956-1957).
The figure formed when the midpoints of adjacent sides of a quadrilateral are joined. Varignon's theorem demonstrated that this figure is a parallelogram. The center of the Varignon parallelogram is the geometric centroid of four point masses placed on the vertices of the quadrilateral.
If a cyclic quadrilateral is inscribed in a circle of a coaxal system such that one pair of connectors touches another circle of the system at , then each pair of opposite connectors will touch a circle of the system ( at on , at on , at on , at on , and at on ), and the six points of contact , , , , , and will be collinear.The general theorem states that if , , ..., are any number of points taken in order on a circle of a given coaxal system so that , , ..., touch respectively fixed circles , , ..., of the system, then must touch a fixed circle of the system. Further, if , , ..., touch respectively any of the circles , , ..., , then must touch the remaining circle.
A quadrilateral in which a pair of opposite sides have the same length and are inclined at to each other (or equivalently, satisfy ). Some interesting theorems hold for such quadrilaterals. Let be an equilic quadrilateral with and . Then 1. The midpoints , , and of the diagonals and the side always determine an equilateral triangle. 2. If equilateral triangle is drawn outwardly on , then is also an equilateral triangle. 3. If equilateral triangles are drawn on , , and away from , then the three new graph vertices , , and are collinear. See Honsberger (1985) for additional theorems.
Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle.van Aubel's theorem is related to Napoleon's theorem and is a special case of the Petr-Neumann-Douglas theorem. It is sometimes (incorrectly) known simply as Aubel's theorem (Casey 1888; Wells 1991, p. 11; Kimberling 2003, p. 23).A second theorem sometimes known as van Aubel's theorem states that if is the Cevian triangle of a point , then
A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. And, since a square is a degenerate case of a rectangle, both squares and rectangles are special types of parallelograms.The polygon diagonals of a parallelogram bisecteach other (Casey 1888, p. 2).The angles of a parallelogram satisfy the identities(1)(2)and(3)A parallelogram of base and height has area(4)The height of a parallelogram is(5)and the polygon diagonals and are(6)(7)(8)(9)(Beyer 1987).The sides , , , and diagonals , of a parallelogram satisfy(10)(Casey 1888, p. 22).The area of the parallelogram with sides formed by the vectors and is(11)(12)(13)where is the two-dimensional cross product and is the determinant.As shown by Euclid, if lines parallel to the sides are drawn through..
A trapezoid is a quadrilateral with two sides parallel. The trapezoid is equivalent to the British definition of trapezium (Bronshtein and Semendyayev 1977, p. 174). An isosceles trapezoid is a trapezoid in which the base angles are equal so . A right trapezoid is a trapezoid having two right angles.The area of the trapezoid is(1)(2)(3)The geometric centroid lies on the median between the base and top, and if the lower left-hand corner of the trapezoid is at the original, lies at(4)(5)(6)(cf. Harris and Stocker 1998, p. 83, who give but not ).The trapezoid depicted has central median(7)If vertical lines are extended from the endpoints of the upper side, the bases of the triangles formed on the left and right are(8)(9)respectively. This gives the vertex angles as(10)(11)(12)(13)from the lower left corner proceeding counterclockwise.In terms of the side length, the diagonals of the trapezoid are given by(14)(15)and the height..
The orthopoles of a line with respect to the four triangles formed by three out of four vertices of any quadrilateral lie on a straight line known as the orthopolar line of for the given quadrilateral (Servais 1923, McBrien 1942, Goormaghtigh 1947).
There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British definition is a quadrilateral with two sides parallel (e.g., Bronshtein and Semendyayev 1977, p. 174)--which Americans call a trapezoid.Definitions for trapezoid and trapezium have causedcontroversy for more than two thousand years.Euclid (Book 1, Definition 22) stated, "Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has opposite sides and angles equal to one another but is neither equilateral nor right angled. And let quadrilaterals other than these be called trapezia."Proclus (also Heron and Posidonius) divided quadrilaterals into parallelograms and non-parallelograms. For the latter, Proclus assigned..
A quadrilateral which has an incircle, i.e., one for which a single circle can be constructed which is tangent to all four sides. Opposite sides of such a quadrilateral satisfy(1)where(2)is the semiperimeter, and the areais(3)where is the inradius. Using Bretschneider's formula together with (1) and (3) then gives the beautiful formula(4)(5)where and are the diagonal lengths.A rhombus is a special case of a tangential quadrilateral.
Given four points, , , , and , let be the orthocenter of . Then is the orthocenter , is the orthocenter of , and is the orthocenter of . The configuration is called an orthocentric quadrangle.
A cyclic quadrilateral is a quadrilateral for which a circle can be circumscribed so that it touches each polygon vertex. A quadrilateral that can be both inscribed and circumscribed on some pair of circles is known as a bicentric quadrilateral.The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths. The opposite angles of a cyclic quadrilateral sum to radians (Euclid, Book III, Proposition 22; Heath 1956; Dunham 1990, p. 121). There exists a closed billiards path inside a cyclic quadrilateral if its circumcenter lies inside the quadrilateral (Wells 1991, p. 11).The area is then given by a special case of Bretschneider's formula. Let the sides have lengths , , , and , let be the semiperimeter(1)and let be the circumradius. Then(2)(3)the first of which is known as Brahmagupta's formula. Solving for the circumradius in (2) and (3) gives(4)The diagonals of a cyclic quadrilateral..
Let , , , and be four points on a circle, and , , , the orthocenters of triangles , etc. If, from the eight points, four with different subscripts are chosen such that three are from one set and the fourth from the other, these points form an orthocentric system. There are eight such systems, which are analogous to the six sets of orthocentric systems obtained using the feet of the angle bisectors, orthocenter, and polygon vertices of a generic triangle.On the other hand, if all the points are chosen from one set, or two from each set, with all different subscripts, the four points lie on a circle. There are four pairs of such circles, and eight points lie by fours on eight equal circles.The Simson line of with regard to triangle is the same as that of with regard to the triangle .
The figure determined by four lines, no three of which are concurrent, and their six points of intersection (Johnson 1929, pp. 61-62). Note that this figure is different from a complete quadrangle. A complete quadrilateral has three diagonals (compared to two for an ordinary quadrilateral). The midpoints of the diagonals of a complete quadrilateral are collinear on a line (Johnson 1929, pp. 152-153).A theorem due to Steiner (Mention 1862ab, Johnson 1929, Steiner 1971) states that in a complete quadrilateral, the bisectors of angles are concurrent at 16 points which are the incenters and excenters of the four triangles. Furthermore, these points are the intersections of two sets of four circles each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the circumcircles of the quadrilateral.Newton proved that, if a conic section is inscribed in a complete quadrilateral,..
The term "square" can be used to mean either a square number (" is the square of ") or a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other as illustrated above. In other words, a square is a regular polygon with four sides.When used as a symbol, denotes a square geometric figure with given vertices, while is sometimes used to denote a graph product (Clark and Suen 2000).A square is a special case of an isosceles trapezoid, kite, parallelogram, quadrilateral, rectangle, rhombus, and trapezoid.The diagonals of a square bisect one another and are perpendicular (illustrated in red in the figure above). In addition, they bisect each pair of opposite angles (illustrated in blue).The perimeter of a square with side length is(1)and the area is(2)The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon..
When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. Furthermore, the area of this parallelogram determined by the edges of lengths and in the figure above is given by(Yetter 1998; Trott 2004, pp. 65-66)
If the four points making up a quadrilateral are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. A complete quadrangle is therefore a set of four points, no three collinear, and the six lines which join them. Note that a complete quadrilateral is different from a complete quadrangle.The midpoints of the sides of any complete quadrangle and the three diagonal points all lie on a conic known as the nine-point conic. If it is an orthocentric quadrilateral, the conic reduces to a circle.
A four-sided quadrilateral not contained in a plane. The lines connecting the midpoints of opposite sides of a skew quadrilateral intersect (and bisect) each other (Steinhaus 1999).The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz (Schwarz 1890, Wells 1991) in terms of Abelian integrals and has the shape of a saddle. It is given by solving
An equilateral parallelogram whose acute angles are . Sometimes, the restriction to is dropped, and it is required only that two opposite angles are acute and the other two obtuse. The term rhombus is commonly used for an arbitrary equilateral parallelogram.The area of a lozenge of side length is(1)its diagonals have lengths(2)(3)and it has inradius(4)
Given a general quadrilateral with sides of lengths , , , and , the area is given by(1)(2)(Coolidge 1939; Ivanov 1960; Beyer 1987, p. 123) where and are the diagonal lengths and is the semiperimeter. While this formula is termed Bretschneider's formula in Ivanoff (1960) and Beyer (1987, p. 123), this appears to be a misnomer. Coolidge (1939) gives the second form of this formula, stating "here is one [formula] which, so far as I can find out, is new," while at the same time crediting Bretschneider (1842) and Strehlke (1842) with "rather clumsy" proofs of the related formula(3)(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123), where and are two opposite angles of the quadrilateral."Bretschneider's formula" can be derived by representing the sides of the quadrilateral by the vectors , , , and arranged such that and the diagonals by the vectors and arranged so that and . The..
A right trapezoid is a trapezoid having two right angles.As illustrated above, the area of a right trapezoid is(1)(2)A right trapezoid has perimeter(3)and diagonal lengths(4)(5)
Pick a point in the interior of a quadrilateral which is not a parallelogram. Join this point to each of the four vertices, then the locus of points for which the sum of opposite triangle areas is half the quadrilateral area is the line joining the midpoints and of the polygon diagonals.
A quadrilateral whose consecutive sides have the lengths , , , , where(1)and(2)Brahmagupta's trapezium is a cyclic quadrilateralwith perpendicular diagonals.It has area(3)circumradius,(4)and the diagonal lengths(5)(6)All these values are rational if and are. In particular, if and are Pythagorean triples, the area, circumdiameter, the lengths of the diagonals are all integers.
A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides the same length, i.e., an equilateral parallelogram. The word rhomb is sometimes used instead of rhombus, and a rhombus is sometimes also called a diamond. A rhombus with is sometimes called a lozenge.The polygon diagonals and of a rhombus are perpendicular and satisfy(1)The diagonals are related to the opening angle by(2)(3)The area of a rhombus is given by(4)(5)(6)The rhombus is a tangential quadrilateral with , and so has inradius(7)(8)
A planar convex quadrilateral consisting of two adjacent sides of length and the other two sides of length . The rhombus is a special case of the kite, and the lozenge is a special case of the rhombus.The area of a kite is given by(1)where(2)(3)are the lengths of the polygon diagonals (whichare perpendicular).The 120-90-60-90 kite with edge ratios is the basis for the polyomino-like objects known as polykites.
In a cyclic quadrilateral having perpendicular diagonals , the perpendiculars to the sides through point of intersection of the diagonals (the anticenter) always bisects the opposite side (so , , , and are the midpoints of the corresponding sides of the quadrilateral).
An isosceles trapezoid (called an isosceles trapezium by the British; Bronshtein and Semendyayev 1997, p. 174) is trapezoid in which the base angles are equal and therefore the left and right side lengths are also equal.From the Pythagorean theorem,(1)so(2)(3)An isosceles trapezoid has perimeter(4)and diagonal lengths(5)
For a general quadrilateral with sides of length , , , and , the area is given by(1)where(2)is the semiperimeter, is the angle between and , and is the angle between and . Brahmagupta's formula(3)is a special case giving the area of a cyclic quadrilateral (i.e., a quadrilateral inscribed in a circle), for which . In terms of the circumradius of a cyclic quadrilateral,(4)The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.For a bicentric quadrilateral (i.e., a quadrilateral that can be inscribed in one circle and circumscribed on another), the area formula simplifies to(5)(6)(Ivanoff 1960; Beyer 1987, p. 124).
A line segment joining the midpointsof opposite sides of a quadrilateral or tetrahedron.Varignon's theorem states that the bimedians of a quadrilateral bisect each other (left figure). In addition, the three bimedians of a tetrahedron are concurrent and bisect each other (right figure; Altshiller-Court 1979, p. 48).
A closed planar quadrilateral with opposite sides of equal lengths and , and with four right angles. A square is a degenerate rectangle with .The area of the rectangle is(1)and its polygon diagonals and are of length(2)A rectangle has a circumcircle with circumradius(3)but incircle only in the degenerate case of a square.A number of important topological surfaces can be constructed from the rectangle. Gluing both pairs of opposite edges together with no twists gives a torus, gluing two opposite edges together after giving a half-twist gives a Möbius strip, gluing both pairs of opposite edges together giving one pair a half-twist gives a Klein bottle, and giving both pairs a half-twist gives a projective plane (Stewart 1997).
A bicentric quadrilateral, also called a cyclic-inscriptable quadrilateral, is a four-sided bicentric polygon. The inradius , circumradius , and offset are connected by the equation(1)(Davis; Durége 1861; Casey 1888, pp. 109-110; Johnson 1929; Dörie 1965; Coolidge 1971, p. 46; Salazar 2006). Finding this relation is sometimes known as Fuss's problem.In addition(2)(3)(Beyer 1987), where is the semiperimeter, and(4)The area of a bicentric quadrilateral is(5)(6)where and are the lengths of the diagonals (Ivanoff 1960; Beyer 1987, p. 124).
A rational quadrilateral is a quadrilateral for which the sides, polygon diagonals, and area are rational. The simplest case has sides , , , and , polygon diagonals of length and , and area 1764.
Given a quadrilateral , the three diagonal points , , and defined as the pairwise intersections of the lines determined by the sides , , and .