Triangle ABC with AB = 4, BC = 5, CA = 6 has circumcircle \Omega and incircle \omega. Let \Gamma be the circle tangent to \Omega and the sides AB, BC, and let X = \Gamma \cap \Omega. Let Y, Z be distinct points on \Omega such that XY, XZ are tangent to \omega. Find YZ^2.
The following fact may be useful: if \triangle ABC has incircle \omega with center I and radius r, and \triangle DEF is the intouch triangle (i.e. D, E, F are intersections of the incircle with BC, CA, AB, respectively) and H is the orthocenter of \triangle DEF, then the inversion of X about \omega (i.e. the point X' on ray IX such that IX' \cdot IX = r^2) is the midpoint of DH.