Quadrilateral whose vertices can all fall on a single circle
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.
A convex quadrilateral ABCD is cyclic if and only if an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal. That is, for example,
Another necessary and sufficient conditions for a convex quadrilateral ABCD to be cyclic are: let E be the point of intersection of the diagonals, let F be the intersection point of the extensions of the sides AD and BC, let be a circle whose diameter is the segment, EF, and let P and Q be Pascal points on sides AB and CD formed by the circle .
(1) ABCD is a cyclic quadrilateral if and only if points P and Q are collinear with the center O, of circle .
(2) ABCD is a cyclic quadrilateral if and only if points P and Q are the midpoints of sides AB and CD.
Intersection of diagonals
If two lines, one containing segment AC and the other containing segment BD, intersect at E, then the four points A, B, C, D are concyclic if and only if
The intersection E may be internal or external to the circle. In the former case, the cyclic quadrilateral is ABCD, and in the latter case, the cyclic quadrilateral is ABDC. When the intersection is internal, the equality states that the product of the segment lengths into which E divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral are chords of the circumcircle.
Ptolemy's theorem expresses the product of the lengths of the two diagonals e and f of a cyclic quadrilateral as equal to the sum of the products of opposite sides:: p.25 
where a, b, c, d are the side lengths in order. The converse is also true. That is, if this equation is satisfied in a convex quadrilateral, then a cyclic quadrilateral is formed.
In a convex quadrilateral ABCD, let EFG be the diagonal triangle of ABCD and let be the nine-point circle of EFG.
ABCD is cyclic if and only if the point of intersection of the bimedians of ABCD belongs to the nine-point circle .
The cyclic quadrilateral has maximal area among all quadrilaterals having the same side lengths (regardless of sequence). This is another corollary to Bretschneider's formula. It can also be proved using calculus.
Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.
The area of a cyclic quadrilateral with successive sides a, b, c, d, angle A between sides a and d, and angle B between sides a and b can be expressed as: p.25
where there is equality if and only if the quadrilateral is a square.
In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as: p.25, : p. 84
In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.
If M and N are the midpoints of the diagonals AC and BD, then
where E and F are the intersection points of the extensions of opposite sides.
If ABCD is a cyclic quadrilateral where AC meets BD at E, then
A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common.: p. 84
Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.: p.131,  These line segments are called the maltitudes, which is an abbreviation for midpoint altitude. Their common point is called the anticenter. It has the property of being the reflection of the circumcenter in the "vertex centroid". Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.
If the diagonals of a cyclic quadrilateral intersect at P, and the midpoints of the diagonals are M and N, then the anticenter of the quadrilateral is the orthocenter of triangleMNP.
The anticenter of a cyclic quadrilateral is the Poncelet point of its vertices.
In a cyclic quadrilateral ABCD, the incentersM1, M2, M3, M4 (see the figure to the right) in trianglesDAB, ABC, BCD, and CDA are the vertices of a rectangle. This is one of the theorems known as the Japanese theorem. The orthocenters of the same four triangles are the vertices of a quadrilateral congruent to ABCD, and the centroids in those four triangles are vertices of another cyclic quadrilateral.
If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.
A Brahmagupta quadrilateral is a cyclic quadrilateral with integer sides, integer diagonals, and integer area. All Brahmagupta quadrilaterals with sides a, b, c, d, diagonals e, f, area K, and circumradius R can be obtained by clearing denominators from the following expressions involving rational parameters t, u, and v:
Circumradius and area
For a cyclic quadrilateral that is also orthodiagonal (has perpendicular diagonals), suppose the intersection of the diagonals divides one diagonal into segments of lengths p1 and p2 and divides the other diagonal into segments of lengths q1 and q2. Then (the first equality is Proposition 11 in Archimedes' Book of Lemmas)
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.
In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.
Cyclic spherical quadrilaterals
In spherical geometry, a spherical quadrilateral formed from four intersecting greater circles is cyclic if and only if the summations of the opposite angles are equal, i.e., α + γ = β + δ for consecutive angles α, β, γ, δ of the quadrilateral. One direction of this theorem was proved by Anders Johan Lexell in 1782. Lexell showed that in a spherical quadrilateral inscribed in a small circle of a sphere the sums of opposite angles are equal, and that in the circumscribed quadrilateral the sums of opposite sides are equal. The first of these theorems is the spherical analogue of a plane theorem, and the second theorem is its dual, that is, the result of interchanging great circles and their poles. Kiper et al. proved a converse of the theorem: If the summations of the opposite sides are equal in a spherical quadrilateral, then there exists an inscribing circle for this quadrilateral.
^ abcdFraivert, David; Sigler, Avi; Stupel, Moshe (2020), "Necessary and sufficient properties for a cyclic quadrilateral", International Journal of Mathematical Education in Science and Technology, 51 (6): 913–938, doi:10.1080/0020739X.2019.1683772, S2CID209930435
^ abcdefgAltshiller-Court, Nathan (2007) , College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), Courier Dover, pp. 131, 137–8, ISBN978-0-486-45805-2, OCLC78063045