Cyclic Quadrilateral - Characterizations

Characterizations

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 its opposite angles are supplementary, that is

The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.

Another necessary and sufficient condition for a convex quadrilateral ABCD to be cyclic is that 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,

Ptolemy's theorem expresses the product of the lengths of the two diagonals p and q of a cyclic quadrilateral as equal to the sum of the products of opposite sides:

The converse is also true. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral.

If two chords of a circle AC and BD intersect at X, then the four points A, B, C, D are concyclic if and only if

The intersection X may be internal or external to the circle. In the former case, the quadrilateral is ABCD, and in the latter case, the convex quadrilateral is ABDC. When the intersection is internal, the equality states that the product of the segment lengths into which X 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.

Yet another characterization is that a convex quadrilateral ABCD is cyclic if and only if