2020 AIME II Problem 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.