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 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 QuadrilateralSpecial Cases, Characterizations, Area, Diagonals, Angle Formulas, Parameshvara's Formula, Anticenter and Collinearities, Other Properties, Brahmagupta Quadrilaterals

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... 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 ...
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... 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 ...
Indian Mathematics - Classical Period (400 – 1200) - Seventh and Eighth Centuries
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