Question 15
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega.$ The tangents to $\omega$ at $B$ and $C$ intersect at $T.$ Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC,$ respectively. Suppose $BT = CT = 16,$ $BC = 22,$ and $TX^2 + TY^2 + XY^2 = 1143.$ Find $XY^2.$