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. The center of the circle and its radius are called the circumcenter and the circumradius respectively.

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 (kuklos), which means "circle" or "wheel".

For Cyclic quadrilateral, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

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.

Structural Signature¶

Sig role-phrases:

Defining carrier — Other necessary and sufficient conditions for a convex quadrilateral to be cyclic are: let be the point of intersection of the diagonals, let be the intersection point of the extensions of the sides and , let be a circle whose diameter is the segment, , and let and be Pascal points on sides and formed by the circle .
Constitutive relation — That is, if this equation is satisfied in a convex quadrilateral, then a cyclic quadrilateral is formed.
Operating condition — The area of a cyclic quadrilateral with sides , , , is given by Brahmagupta's formula.
Recognition evidence — Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area.
Admissible variation — A generalization of Mollweide's formula to cyclic quadrilaterals is given by the following two identities.
Characteristic consequence — This was derived by the Indian mathematician Vatasseri Parameshvara in the 15^th century.
Failure boundary — Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by 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.
Not an over-broad reading. Provided is not a right angle, the area can also be expressed as.
Not an over-broad reading. See for a different parameterization of all non-degenerate primitive Brahmagupta quadrilaterals, which depends upon rational numbers, .
Not an over-broad reading. All triangles have a circumcircle, but not all quadrilaterals do.
Not automatically Tangential quadrilateral. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Cyclic quadrilateral applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

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.
Supplementary angles. A convex quadrilateral is cyclic if and only if its opposite angles are supplementary, that is.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Provided is not a right angle, the area can also be expressed as. so that a reader can reproduce the classification rather than infer it from topical resemblance.

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. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

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. Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area.
Test variation. Change an implementation or setting while preserving a generalization of Mollweide's formula to cyclic quadrilaterals is given by the following two identities.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In the former case, the cyclic quadrilateral is , and in the latter case, the cyclic quadrilateral is . This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area

Applied / In Practice¶

This is a corollary of Bretschneider's formula for the general quadrilateral, since opposite angles are supplementary in the cyclic case. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Area; invariant → 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; boundary → the case exits the class when provided is not a right angle, the area can also be expressed as

Structural Tensions¶

T1 — Stable identity versus admissible variation. Provided is not a right angle, the area can also be expressed as. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. See for a different parameterization of all non-degenerate primitive Brahmagupta quadrilaterals, which depends upon rational numbers, . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. All triangles have a circumcircle, but not all quadrilaterals do. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. A kite is cyclic if and only if it has two right angles – a right kite. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Other necessary and sufficient conditions for a convex quadrilateral to be cyclic are: let be the point of intersection of the diagonals, let be the intersection point of the extensions of the sides and , let be a circle whose diameter is the segment, , and let and be Pascal points on sides and formed by the circle . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Cyclic quadrilateral literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. That is, if this equation is satisfied in a convex quadrilateral, then a cyclic quadrilateral is formed. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Cyclic quadrilateral distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Cyclic quadrilateral is structural-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The area of a cyclic quadrilateral with sides , , , is given by Brahmagupta's formula. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Other necessary and sufficient conditions for a convex quadrilateral to be cyclic are: let be the point of intersection of the diagonals, let be the intersection point of the extensions of the sides and , let be a circle whose diameter is the segment, , and let and be Pascal points on sides and formed by the circle . That is, if this equation is satisfied in a convex quadrilateral, then a cyclic quadrilateral is formed. It further constrains recognition and variation through: The area of a cyclic quadrilateral with sides , , , is given by Brahmagupta's formula. Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Cyclic quadrilateral literal. Its documented scope includes the condition that For a cyclic quadrilateral with successive sides , , , , semiperimeter , and angle between sides and , the trigonometric functions of are given by. Another bounded application condition is that A kite is cyclic if and only if it has two right angles – a right kite. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—A generalization of Mollweide's formula to cyclic quadrilaterals is given by the following two identities.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Polygon.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Cyclic quadrilateral. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

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

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish 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?
Tangential quadrilateral. A convex quadrilateral whose four sides are tangent to one inscribed circle. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Law of sines. Relate each side of a triangle to the sine of its opposite angle through one common ratio equal to the circumdiameter in Euclidean geometry, enabling triangle solution while preserving the side-side-angle ambiguous case and curvature-specific variants. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Rhombus. A nondegenerate Euclidean quadrilateral with four equal sides, forcing a parallelogram whose diagonals bisect at right angles and whose square case adds right-angle symmetry. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Cyclic quadrilateral remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Cyclic_quadrilateral (revision 1359811160).
Preserved source candidate: https://books.google.com/books?id=ZkoUR5lRwdcC&pg=PA63
Preserved source candidate: http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII22.html
Preserved source candidate: http://forumgeom.fau.edu/FG2008volume8/FG200814.pdf
Preserved source candidate: https://web.archive.org/web/20191126135341/http://forumgeom.fau.edu/FG2008volume8/FG200814.pdf
Preserved source candidate: https://books.google.com/books?id=mwUHJpvLOPsC&pg=PA44
Preserved source candidate: https://archive.org/details/mathematicalolym0000andr_q5g0/page/44
Preserved source candidate: https://ijgeometry.com/wp-content/uploads/2018/04/5-16.pdf
Preserved source candidate: http://students.imsa.edu/~tliu/Math/planegeo.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.