Mock AIME 3 Pre 2005 Problems/Problem 7
Problem
is a cyclic quadrilateral that has an inscribed circle. The diagonals of
intersect at
. If
and
then the area of the inscribed circle of
can be expressed as
, where
and
are relatively prime positive integers. Determine
.
Solution
Let and
. Angle-chasing can be used to prove that
. Therefore
. This shows that
and
. More angle-chasing can be used to prove that
. This shows that
. It is a well-known fact that if
is circumscriptable around a circle then
. Therefore
. We also know that
, so we can solve (algebraically or by inspection) to get that
and
.
Heron's Formula states that the area of a cyclic quadrilateral is , where
is the semiperimeter and
,
,
, and
are the side lengths of the quadrilateral. Therefore the area of
is
. It is also a well-known fact that the area of a circumscriptable quadrilateral is
, where
is the inradius. Therefore
. Therefore the area of the inscribed circle is
, and
.