Bicentric quadrilateral
type of shape
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.
If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a special case of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867).
Special cases
Examples of bicentric quadrilaterals are squares, right kites, and isosceles tangential trapezoids.
Characterizations
A convex quadrilateral ABCD with sides a, b, c, d is bicentric if and only if opposite sides satisfy Pitot's theorem for tangential quadrilaterals and the cyclic quadrilateral property that opposite angles are supplementary; that is,
{
a
+
c
=
b
+
d
A
+
C
=
B
+
D
=
π
.
{\displaystyle {\begin{cases}a+c=b+d\\A+C=B+D=\pi .\end{cases}}}
Three other characterizations concern the points where the incircle in a tangential quadrilateral is tangent to the sides.
The public source identifies “Bicentric quadrilateral” as type of shape. This brief keeps that definition visible, then builds a research path around Bicentric, quadrilateral and type.
Why this record matters
A short description can identify a subject without explaining its stakes. For “Bicentric quadrilateral”, the useful work is to connect “type of shape” to the records capable of establishing context and consequence.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Aug 19, 2026. The linked authority identifier is Q282952. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1788 and 1867.
Overview language is designed for orientation and should not be treated as a substitute for the evidence cited beneath it. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Bicentric quadrilateral”, its source revision and the description used here.
- Expand the search: follow Bicentric quadrilateral primary sources, Bicentric quadrilateral archive and Bicentric research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Bicentric quadrilateral”?
- Which institution is responsible for the underlying evidence?
- What terminology or title could unlock a more precise catalogue search?
Search terms from this dossier
This entry incorporates text from “Bicentric quadrilateral” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.