Let ABCD be a cyclic quadrilateral and M,N be the midpoints of AB, CD respectively. The diagonals AC and BD intersect at L. Suppose that the circumcircle of LMN, with center T, intersects the circumcircle of ABCD at two distinct points X,Y. If the line MN intersects the line XY at S and the line XM intersects the line YN at P, prove that PL is perpendicular to ST. perpendicularcyclic quadrilateralgeometry