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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 7-8, Math Kangaroo 9-10.

It is not always possible to inscribe a circle in a quadrilateral. This is because it is not guaranteed that the interior angle bisectors of a quadrilateral intesect at one point. Remember there is a theorem that guarantees this for triangles. However, the interior angle bisectors of quadrilaterals may look like this:

figure

A quadrilateral in which we can inscribe a circle must have interior angle bisectors that intersect at one point:

figure

It is easy to prove that the quadrilaterals OPBQ, OQCM,OMDN,ONAP are kites.

figure

The condition for the quadrilateral to be circumscribable is:

equation