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## 'Bicentric Quadrilaterals' printed from http://nrich.maths.org/

First, try the problem
Circles in Quadrilaterals to
familiarise yourself with the properties of tangential
quadrilaterals.
A bicentric quadrilateral is both tangential and cyclic. In other
words, it is possible to draw a circle inside it which touches all
four sides, and also to draw another circle around it which passes
through all four vertices. (The two circles do not necessarily have
the same centre!)

Here is a picture of a bicentric quadrilateral.

Think about special quadrilaterals,
such as squares, trapezia, and parallelograms.

- Which types of quadrilateral are
always bicentric?
- Which types of quadrilateral are
never bicentric?
- For the quadrilaterals which are
sometimes bicentric, can
you explain the conditions necessary for them to be bicentric?

There is a formula for finding the area $A$ of a bicentric
quadrilateral:

$$ A = \sqrt{abcd} $$

where $a,b,c$ and $d$ are the lengths of the four sides.

Verify that this formula gives the correct area for the
examples of bicentric quadrilaterals you have found.