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
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
In general, there are three essentially distinct cyclic quadrilaterals (modulo rotation and reflection) whose edges are permutations of the lengths 
, 
, 
, and 
. Of the six corresponding polygon diagonals lengths, three are distinct. In addition to 
and 
, there is therefore a "third" polygon diagonal which can be denoted 
. It is given by the equation
An application of Brahmagupta's theorem gives the pretty result that, for a cyclic quadrilateral with perpendicular diagonals, the distance from the circumcenter
to a side is half the length of the opposite side
, , , and . Of the six corresponding polygon diagonals lengths, three are distinct. In addition to and , there is therefore a "third" polygon diagonal which can be denoted . It is given by the equation An application of Brahmagupta's theorem gives the pretty result that, for a cyclic quadrilateral with perpendicular diagonals, the distance from the circumcenter
to a side is half the length of the opposite side
Weisstein, Eric W. "Cyclic Quadrilateral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CyclicQuadrilateral.html
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