Cyclic quadrilateral

The word cyclic is from the Ancient Greek κύκλος (kuklos), which means "circle" or "wheel".

The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.

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

[1] A convex quadrilateral ABCD is cyclic if and only if its opposite angles are supplementary, that is[1][2] The direct theorem was Proposition 22 in Book 3 of Euclid's Elements.

In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2n-gon, then the two sums of alternate interior angles are each equal to (n-1)

[7] That is, for example, Other 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

(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.

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:[9]: p.25 [2] where a, b, c, d are the side lengths in order.

[10][11][2] The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula[9]: p.24 where s, the semiperimeter, is s = ⁠1/2⁠(a + b + c + d).

This is a corollary of Bretschneider's formula for the general quadrilateral, since opposite angles are supplementary in the cyclic case.

[12] Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals,[13] which by Brahmagupta's formula all have the same area.

Provided A is not a right angle, the area can also be expressed as[9]: p.26 Another formula is[14]: p.83 where R is the radius of the circumcircle.

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[9]: p.25,  [16][17]: p. 84 so showing Ptolemy's theorem According to Ptolemy's second theorem,[9]: p.25,  [16] using the same notations as above.

If ABCD is a cyclic quadrilateral where AC meets BD at E, then[19] 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).

, then:[21] A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has the circumradius (the radius of the circumcircle) given by[16][22] This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.

[23]: p.131,  [24] These line segments are called the maltitudes,[25] which is an abbreviation for midpoint altitude.

Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.

Then[28] (the first equality is Proposition 11 in Archimedes' Book of Lemmas) where D is the diameter of the circumcircle.

These equations imply that the circumradius R can be expressed as or, in terms of the sides of the quadrilateral, as[23] It also follows that[23] Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as A formula for the area K of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining Ptolemy's theorem and the formula for the area of an orthodiagonal quadrilateral.

The result is[29]: p.222 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.

[31] 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.