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 are valid in the convex case.
The word cyclic is from the Greek kuklos which means "circle" or "wheel".
All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.
Read more about Cyclic Quadrilateral: Special Cases, Characterizations, Area, Diagonals, Angle Formulas, Parameshvara's Formula, Anticenter and Collinearities, Other Properties, Brahmagupta Quadrilaterals
Other articles related to "cyclic quadrilateral, cyclic, quadrilateral":
... If four points A, B, C, D are given that form a cyclic quadrilateral, then the nine-point circles of ABC, BCD, CDA and DAB concur at the anticenter of the cyclic ... circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle ... circles that is centered at the anticenter of the cyclic quadrilateral ...
... Aditya expressed characteristics of a cyclic quadrilateral, like Brahmagupta did previously ... Mahavira also established equations for the sides and diagonal of Cyclic Quadrilateral ... If sides of Cyclic Quadrilateral are a, b, c, d and its diagonals are x and y while ...
... In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect ... Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side ... If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side ...
... he stated his famous theorem on the diagonals of a cyclic quadrilateral Brahmagupta's theorem If a cyclic quadrilateral has diagonals that are perpendicular to each other, then ... also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e ... Brahmagupta's formula The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by where s, the semiperimeter, given by ...