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Bicentric quadrilateral
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[1] and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral[2] and double scribed quadrilateral.[3]
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.[4] This is a special case of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867).
- ^ Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover. pp. 188–193. ISBN 978-0-486-61348-2.
- ^ Cite error: The named reference
yunwas invoked but never defined (see the help page). - ^ Leng, Gangsong (2016). Geometric Inequalities: In Mathematical Olympiad and Competitions. Shanghai: East China Normal University Press. p. 22. ISBN 978-981-4704-13-7.
- ^ Weisstein, Eric W. "Poncelet Transverse." From MathWorld – A Wolfram Web Resource, [1]
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رباعي أضلاع ثنائي المركز Arabic Cuadrilátero bicéntrico Spanish Bisentrinen nelikulmio Finnish Quadrilatère bicentrique French Bicentrikus négyszög Hungarian 双心四角形 Japanese Двоцентричен четириаголник MK Patrulater bicentric Romanian Вписанно-описанный четырёхугольник Russian Bicentrični štirikotnik SL
