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 π 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. The area is then given by a special case of Bretschneider's formula.
bicentric quadrilateral | Brahmagupta's theorem | Brahmagupta's trapezium | Bretschneider's formula | butterfly theorem | concyclic | cyclic polygon | cyclic quadrangle | Euler brick | Heron's formula | maltitude | mid-arc points | nine-point center | orthocenter | Poncelet transverse | Ptolemy's theorem | quadrilateral | tangential quadrilateral | triangle centroid
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