All triangles and all regular polygons are bicentric.
2.
For a tangential quadrilateral with given sides, the inradius is cyclic ( and hence a bicentric quadrilateral ).
3.
For the circumradius-to-inradius ratio for various " n ", see Bicentric polygon # Regular polygons.
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For an ex-tangential quadrilateral with given sides, the exradius is cyclic ( and hence an ex-bicentric quadrilateral ).
5.
This strict definition excludes pairs of bicentric points such as the Brocard points ( which are interchanged by a mirror-image reflection ).
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On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.
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The four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius.
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Note that the converse does not hold : Some quadrilaterals that are not bicentric also have area \ displaystyle K = \ sqrt { abcd }.
9.
Several binary operations, such as midpoint and trilinear product, when applied to the two Brocard points, as well as other bicentric pairs, produce triangle centers.
10.
If a polygon is both tangential and cyclic, it is called bicentric . ( All triangles are bicentric, for example . ) The incentre and circumcentre of a bicentric polygon are not in general the same point.
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