Cyclic quadrilateral¶

In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8826
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Euclidean Geometry → Mathematics

Core Idea¶

Cyclic quadrilateral is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.

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Corners on a Circle

Draw a circle. Now put four dots on the circle's edge and connect them with straight lines to make a four-sided shape. Because all four corners sit right on the circle, that shape is called a cyclic quadrilateral.

Corners-on-a-Circle Shape

A cyclic quadrilateral is a four-sided shape whose four corners all lie on one circle. Each side is then a straight line connecting two points of the circle, called a chord. The circle is called the circumcircle, its center the circumcenter, and its radius the circumradius. Not every four-sided shape can do this: you can always fit a circle through three points, but a fourth corner has to land exactly on that same circle.

Inscribed Quadrilateral

A cyclic (or inscribed) quadrilateral is a four-sided polygon whose four vertices all lie on one circle. That circle is its circumcircle; its center and radius are the circumcenter and circumradius, and the vertices are called concyclic. Because each vertex is on the circle, every side is a chord of the circle. Any three points not on a line determine a circle, so the real condition is that the fourth vertex lands on the circle through the other three. Usually these quadrilaterals are assumed convex, though there are also crossed versions where two sides cross. The name comes from the Greek word kuklos, meaning circle or wheel.

A cyclic (inscribed) quadrilateral is a quadrilateral whose four vertices are concyclic—lying on a single circle, the circumcircle, with center the circumcenter and radius the circumradius—so that each side is a chord of that circle. Being cyclic is a genuine constraint: three non-collinear points determine a unique circle, and the quadrilateral is cyclic exactly when the fourth vertex lies on that circle. By default the quadrilateral is taken to be convex, and the standard formulas and properties are stated for the convex case; crossed (self-intersecting) cyclic quadrilaterals also exist but need separate treatment. The name derives from the Ancient Greek kuklos, "circle" or "wheel." The identifying condition is the concyclicity of all four vertices; merely resembling a familiar example is not enough.

Scope of Application¶

Angle formulas. For a cyclic quadrilateral with successive sides , , , , semiperimeter , and angle between sides and , the trigonometric functions of are given by.
Special cases. A kite is cyclic if and only if it has two right angles – a right kite.
Special cases. 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.
Special cases. A harmonic quadrilateral is a cyclic quadrilateral in which the product of the lengths of opposite sides are equal.
CharacterizationsCircumcenter. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent.

Clarity¶

A clear use of Cyclic quadrilateral names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle.

Manages Complexity¶

Cyclic quadrilateral compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—that is, if this equation is satisfied in a convex quadrilateral, then a cyclic quadrilateral is formed.—and the practical consequence—this was derived by the Indian mathematician Vatasseri Parameshvara in the 15^th century. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle.
Check operation and conditions. The area of a cyclic quadrilateral with sides , , , is given by Brahmagupta's formula.
Demand recognition evidence.

Knowledge Transfer¶

Within the home domain. Knowledge about Cyclic quadrilateral transfers literally when a new case preserves the same carrier type, relation, and recognition test. For a cyclic quadrilateral with successive sides , , , , semiperimeter , and angle between sides and , the trigonometric functions of are given by. A kite is cyclic if and only if it has two right angles – a right kite. Beyond the home domain. No canonical parent is asserted for Cyclic quadrilateral.

Relationships to Other Abstractions¶

Current abstraction Cyclic quadrilateral Domain-specific

Parents (1) — more general patterns this builds on

Cyclic quadrilateral is a kind of Polygon Domain-specific

It is a four-sided polygon whose vertices lie on a circle.

Hierarchy path (1) — routes to 1 parentless root

Cyclic quadrilateral → Polygon

Neighborhood in Abstraction Space¶

Cyclic quadrilateral sits in a moderately populated region (55^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08