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From Wikipedia
In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5sided), hexagon (6sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
Quadrilaterals are simple (not selfintersecting) or complex (selfintersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple quadrilateral add up to 360 degrees of arc. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720Â°.
All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.
Convex quadrilaterals  parallelograms
A parallelogram is 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 also include the square, rectangle, rhombus and rhomboid.
 Rhombus or rhomb: all four sides are of equal length. Equivalent conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. An informal description is "a pushedover square" (including a square).
 Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushedover rectangle with no right angles."
 Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. 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), 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.
 Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).
A shape that is both a rhombus (four equal sides) and a rectangle (four equal angles) is a square (four equal sides and four equal angles).
Convex quadrilaterals  other
 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. (It is common, especially in the discussions on plane tessellations, to refer to the concave quadrilateral with these properties as a dart or arrowhead, with term kite being restricted to the convex shape.)
 Orthodiagonal quadrilateral: the diagonals cross at right angles.
 Trapezium (British English) or trapezoid (NAm.): one pair of opposite sides are parallel.
 Isosceles trapezium (Brit.) or isosceles trapezoid (NAm.): one pair of opposite sides are parallel and the base angles are equal in measure. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is: "a quadrilateral with an axis of symmetry bisecting one pair of opposite sides".
 Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
 Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A quadrilateral is cyclic if and only if opposite angles sum to 180Â°.
 Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
 Bicentric quadrilateral: both cyclic and tangential.
Area of a convex quadrilateral
There are various general formulas for the area of a convex quadrilateral.
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
 \frac{1}{2} {AC}\times{BD},
which is the magnitude of the cross product of vectors AC and BD. In twodimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x_{1},y_{1}) and BD as (x_{2},y_{2}), this can be rewritten as:
In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The vertices are said to be concyclic.
In a cyclic simple (nonselfintersecting) quadrilateral, opposite angles are supplementary (their sum is Ï€ radians or 180Â°). Equivalently, each exterior angle is equal to the opposite interior angle.
Area
The area of a cyclic quadrilateral is given by Brahmagupta's formula as long as the sides are given:
 \sqrt{(sa)(sb)(sc)(sd)}
where s, the semiperimeter, is s=\frac{a+b+c+d}{2}.
This area is maximal among all quadrilaterals having the same sequence of side lengths.
The area of a cyclic quadrilateral with successive sides a, b, c, d and angle \gamma between sides b and c can also be expressed as
 Area=\tfrac{1}{2}(\sin \gamma)(bc+ad).
Diagonals
Ptolemy's theorem expresses the product of the lengths of the two diagonals p and q of a cyclic quadrilateral as equal to the sum of the products ac and bd of opposite sides:
 pq = ac + bd.
In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.
A cyclic quadrilateral with successive vertices A, B, C, D and successive sides a=AB, b=BC, c=CD, and d=DA and with diagonals p=AC and q=BD has:
 \tfrac {p}{q}= \tfrac{ad+cb}{ab+cd},
 p^{2}= \tfrac{(ac+bd)(ad+bc)}{ab+cd},
and
 q^{2}= \tfrac{(ac+bd)(ab+dc)}{ad+bc}.
If the intersection of the diagonals divides one diagonal into segments of lengths e and f, and divides the other diagonal into segments of lengths g and h, then ef = gh. (This holds because both diagonals are chords of a circle.)
Special cases
Any square, rectangle, or isosceles trapezoid is cyclic. A kite is cyclic if and only if it has two right angles.
Other properties
A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has circumradius (the radius of the circumscribing circle) given by
 \tfrac{1}{4} \sqrt{\tfrac{(ab+cd)(ac+bd)(ad+bc)}{(sa)(sb)(sc)(sd)}}.
There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.
For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d, the trigonometric functions of A are given by
 \cos A = \frac{a^2 + d^2  b^2  c^2}{2(ad + bc)};
 \sin^2 A = \frac{4(sa)(sb)(sc)(sd)}{(ad+bc)^2};
 \tan \frac{A}{2} = \sqrt{\frac{(sa)(sd)}{(sb)(sc)}}.
Four lines, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.
Properties of cyclic quadrilaterals that are also orthodiagonal
Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal (has mutually perpendicular diagonals), the perpendicular from any side through the point of intersection of the diagonals bisects the other side.
If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side.
For a cyclic orthodiagonal quadrilateral, suppose the intersection of the diagonals divides one diagonal into segments of lengths p_{1} and p_{2} and divides the other diagonal into segments of lengths q_{1} and q_{2}. Then
where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. Equivalently, letting R = D / 2 be the radius of the circumcircle, the average of p_1^2, p_2^2, q_1^2, and q_2^2 is R^2. Moreover, the equations a^{2} + c^{2} = b^{2} + d^{2} = D^{2} imply that in an orthodiagonal cyclic quadrilateral, the sum of the squares of the sides equals eight times the square of the circumradius.
 Cyclic polygon
 Ptolemy's theorem
 Brahmagupta's theorem on perpendicular diagonals of cyclic quadrilaterals
 Orthodiagonal quadrilateral
 Tangential quadrilateral, a quadrilateral all of whose sides are tangent to a single circle
 Japanese theorem for cyclic quadrilaterals
From Yahoo Answers
Answers:huh? a quadrilateral is any 4 sided shape in the world that is encloses an area. parallelogram is a particular 4 sided shape that has 2 different sets of parallel sides that are differing lenghts that are also paralel to each other. see here for help: http://www.mathopenref.com/parallelogram.html
Answers:congruent are exactly the same size and shape but similiar have just the same shape or just the same size. but i dont know about the symbols. srry!
Answers:A quadrilateral with both inscribed and circumscribed circles is called "bicentric". The problem of finding the distance between the centers is known as Fuss' Problem and the solution "x" (less than R) satisfies: 1/Ri = 1/(Rc+x) + 1/(Rcx) x = [Rc + Ri  Ri (4Rc + Ri )] However, I don't have a proof short enough to present here. See the reference below. To find Rc and Ri in terms of the sides, it helps to have some area formulas. The trig triangle area formula {Area=(1/2)abSin( )} gives two different formulas (splitting the rectangle into triangles with the two diagonals, subtending angles and ). There is also the Brahmagupta formula using the semiperimeter s=a+c=b+d. (See the reference belwo for proof.) Area = (1/2)(ab+cd)Sin( ) = (1/2)(ad+bc)Sin( ) = [(sa)(sb)(sc)(sd)] = [abcd] Also, splitting the rectangle into four triangles by connecting the incenter with the vertices produces: Area = (1/2)a(Ri) + (1/2)b(Ri) + (1/2)c(Ri) + (1/2)c(Ri) = Ri=2 [abcd]/(a+b+c+d) Finally Ptolemy (proof in the references), letting p and q be the diagonals of the rectangle, which are bases of isosceles triangles with sides (Rc) and central angles 2 and 2 ac+bd = pq = [2(Rc)Sin( )][2(Rc)Sin( )) (Rc) = (ac+bd) / (4Sin Sin ) Multiplying the top and bottom by Area : (Rc) = (ac+bd) / (4Sin Sin ) (1/2)(ab+cd)Sin( ) (1/2)(ad+bc)Sin( ) / (abcd) Producing: Rc = (1/4) [(ac+bd)(ab+cd)(ad+bc)/(abcd)] We could substitute this expression for Rc along with Ri=2 [abcd]/(a+b+c+d) into the above formula for x= [Rc +Ri Ri (4Rc +Ri )], but it doesn't look like it will simplify nicely and I'm too lazy to try.
Answers:Hi, A quadrilateral is a 4sided figure, but it doesn't have to be a rectangle, square or parallelogram. Instead it can be a Vshaped concave figure and still have 4 sides. With this type of quadrilateral, a triangle can intersect with it 8 times.<==ANSWER . . . . . . . . . . . ./ . . . . . . / . . . . . . . ./ . . . . . . . . . / .. . . . . . . . . . ./ . . . . . . . . . . TRIANGLE . . . / . . . . . . . . . . . . . \ /. . . . . . . . . . . . . . . . . . . / \ ./ . . . . . . . . . . . . . . . . ./ . \/ . . . . . . . . . . . . . . . . . / . / \ . . . . . . . . . . . . . . . . . ./ . /___\______ _______. ____./____ BASE of TRIANGLE . . . . \ . . . . . . . . . . . . . . . / . . . . . \ . . . . . . . . . . . . . / . . . . . . \ . . . . . . . * . . . . . / . . . . . . . \ . . . . . . . . . . . ./ . . . . . . . .\ . . . . . . . . . . ./ . . . . . . . . \ . . . . . . . . . ./ Vshaped quadrilateral . . . . . . . . .\ . . . . . . . . ./ . . . . . . . . . \ . . . . . . . ./ . . . . . . . . . . \. . . . . . ./ . . . . . . . . . . .\ . . . . ./ . . . . . . . . . . . \ . . . ./ . . . . . . . . . . . .\. . ./ . . . . . . . . . . . . \. ./ . . . . . . . . . . . . .\/ I hope that helps!! :)
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