Cyclic quadrilateral

A quadrilateral whose four vertices lie on a single circle. Its opposite interior angles are supplementary, and its area can be found from its side lengths alone.

Opposite angles of a cyclic quadrilateral: The opposite interior angles of a cyclic quadrilateral sum to 180 degrees. This is the defining angle relation produced by four vertices on one circle.
Quadrilateral: A polygon with four sides and four vertices. Cyclicity adds a circle condition to this broader polygon class.
Tangential quadrilateral: A quadrilateral with a circle tangent to all four of its sides. Cyclicity constrains vertices to a circle, while tangentiality constrains sides.
Concyclic points: Points that lie on a common circle. A cyclic quadrilateral is the four-point case of concyclicity.
Ptolemy's theorem: For a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides. It links all four side lengths to both diagonals when the vertices are concyclic.
Circle: The set of points in a plane at a fixed distance from a center. All four vertices must lie on this geometric locus.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel. A parallelogram is cyclic only when it is a rectangle.
Cyclic polygon: A polygon whose vertices all lie on one circle. It generalizes the quadrilateral's defining condition to any number of vertices.
Inscribed angle theorem: An angle subtended by a circle's arc at its circumference is half the arc's central angle. It explains why opposite angles intercept arcs that together make a full circle.
Inscribed angle: An angle whose vertex lies on a circle and whose sides intersect the circle. Angles at the quadrilateral's vertices are inscribed angles of its circumcircle.

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