AIME II 2010 #15

[stupidityismygam]

In triangle [unparseable or potentially dangerous latex formula]. Points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] lie on [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. Points [unparseable or potentially dangerous latex formula] lie of [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula]. Let [unparseable or potentially dangerous latex formula] be the point, other than [unparseable or potentially dangerous latex formula], of intersection of the circumcircles of [unparseable or potentially dangerous latex formula]. Ray [unparseable or potentially dangerous latex formula] meets [unparseable or potentially dangerous latex formula] at [unparseable or potentially dangerous latex formula]. The ratio [unparseable or potentially dangerous latex formula] can be written in the form [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers. Find [unparseable or potentially dangerous latex formula]