Mock AIME 3 Pre 2005 Problems/Problem 7
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 .
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 .
Brahmagupta'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 .