AIME I 2009 #15

[stupidityismygam]

In triangle [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] be a point in the interior of [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] denote the incenters of triangles [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] respectively. The circumcircles of triangles [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] meet at distinct points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] The maximum possible area of [unparseable or potentially dangerous latex formula] can be expressed in the form [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are positive integers and [unparseable or potentially dangerous latex formula] is not divisible by the square of any prime. Find [unparseable or potentially dangerous latex formula]