Projections of a Tangent Intersection

Let $\omega$ be the circumcircle of an acute scalene triangle $ABC$. The tangents to $\omega$ at $B$ and $C$ meet at a point $T$, and $X,Y$ are the feet of the perpendiculars from $T$ to the lines $AB,AC$. Given $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$, find $XY^2$.