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Bicentric Quadrilaterals
Age 14 to 16
Challenge Level 
Why do this problem
This problem provides an opportunity to consider properties of quadrilaterals and circle theorems while investigating the unfamiliar and intriguing idea of bicentric quadrilaterals.Possible approach
First, try the problem Circles
in Quadrilaterals. For learners who are unfamiliar with the
properties of cyclic quadrilaterals, the problems Triangles
in Circles and Pegboard
Quads can be used to investigate.
Explain that a bicentric quadrilateral is both cyclic and
tangential, so a circle can be drawn inside it just touching each
side, and another circle (not necessarily with the same centre) can
be drawn around it just touching each vertex. Ask learners to
sketch examples of bicentric quadrilaterals, and encourage them to
share ideas in pairs about how to identify which quadrilaterals are
bicentric.
Bring the class together for discussion of their ideas. One
hint for finding bicentric quadrilaterals is to start with a
tangential one and make it also cyclic, or vice versa.
Experimenting with a dynamic geometry package such as Geogebra can give some insight
into the properties of these quadrilaterals.
Then give pairs or small groups time to construct some
examples of bicentric quadrilaterals and calculate their areas. The
areas can then be used to verify the area formula given for these
cases.
Key questions
What properties must a quadrilateral have to be
tangential?
What properties must a quadrilateral have to be cyclic?
Which quadrilaterals can be both tangential and cyclic at the
same time?
Possible extension
Show that the area formula given will hold for specific types of quadrilateral such as squares and kites, given the constraint that they are bicentric. Proving the formula generally is extremely challenging!Possible support
Try Triangles
in Circles as a basis for investigating circle theorems.
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