USAMO 2006 Problem 6

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$ , respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$ . Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ , respectively. Prove that the circumcircles of triangles $SAE$ , $SBF$ , $TCF$ , and $TDE$ pass through a common point.