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 (Wells 1991, p. 11).
The area is then given by a special case of Bretschneider's formula. Let the sides have lengths 
, 
, 
, and 
, let 
be the semiperimeter
 |
(1)
|
and let 
be the circumradius. Then
 |  |  |
(2)
|
 |  |  |
(3)
|
the first of which is known as Brahmagupta's formula. Solving for the circumradius in (2) and (3) gives
 |
(4)
|
The diagonals of a cyclic quadrilateral have lengths
 |  |  |
(5)
|
 |  |  |
(6)
|
so that 
.
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
 |
(7)
|
This allows the area formula to be written in the particularly beautiful and simple form
 |
(8)
|
The polygon diagonals are sometimes also denoted 
,
,
and 
.
|
|
The incenters of the four triangles composing the cyclic quadrilateral form a rectangle. Furthermore, the sides
of the rectangle are parallel
to the lines connecting the mid-arc points between
each pair of vertices (left figure above; Fuhrmann 1890, p. 50; Johnson 1929,
pp. 254-255; Wells 1991). If the excenters of the
triangles constituting the quadrilateral are added to the incenters,
a 
rectangular grid is obtained (right figure; Johnson 1929, p. 255; Wells 1991).
Consider again the four triangles contained in a cyclic quadrilateral. Amazingly, the triangle centroids 
, nine-point centers 
, and orthocenters 
formed by these triangles are similar
to the original quadrilateral. In fact, the triangle formed by the orthocenters
is congruent to it (Wells 1991, p. 44).
A cyclic quadrilateral with rational sides 
, 
, 
, and 
, polygon diagonals 
and 
, circumradius 
, and area 
is given by 
, 
, 
, 
, 
, 
, 
, and 
.
Let 
be a quadrilateral such that the angles 
and 
are right angles, then
is a cyclic quadrilateral (Dunham 1990). This is a corollary
of the theorem that, in a right triangle, the midpoint of the hypotenuse
is equidistant from the three vertices. Since 
is the midpoint of both right
triangles 
and 
,
it is equidistant from all four vertices, so a circle
centered at 
may be drawn through them. This theorem is one of the building blocks of Heron's
derivation of Heron's formula.
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, so in the
above figure,
 |
(9)
|
and so on (Honsberger 1995, pp. 37-38).
Let 
and 
be the midpoints of the diagonals of a cyclic quadrilateral
,
and let 
be the intersection of the diagonals. Then the orthocenter
of triangle 
is the anticenter 
of 
(Honsberger 1995, p. 39).
Place four equal circles so that they intersect in a point. The quadrilateral 
is then a cyclic quadrilateral (Honsberger 1991). For a
convex cyclic quadrilateral 
, consider the set of convex cyclic
quadrilaterals 
whose sides are parallel to 
. Then the 
of maximal area is the one
whose polygon diagonals are perpendicular
(Gürel 1996).
