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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. The word cyclic is from the Ancient Greek [[wiktionary:κύκλος#Ancient 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.

Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles – a right kite. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex bicentric quadrilateral is a cyclic quadrilateral that is also ex tangential. A harmonic quadrilateral is a cyclic quadrilateral in which the product of the lengths of opposite sides are equal.

A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.

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. 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.

Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent. 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 triangle MNP. The anticenter of a cyclic quadrilateral is the Poncelet point of its vertices.

In a cyclic quadrilateral ABCD, the incenters M1, M2, M3, M4 (see the figure to the right) in triangles DAB, 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 a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex bicentric. 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:

In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect. 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.