Let \(ABC\) be an acute-angled triangle with \(AC > AB\), let \(O\) be its circumcentre, and let \(D\) be a point on the segment \(BC\). The line through \(D\) perpendicular to \(BC\) intersects the lines \(AO, AC,\) and \(AB\) at \(W, X,\) and \(Y,\) respectively. The circumcircles of triangles \(AXY\) and \(ABC\) intersect again at \(Z \ne A\).

Prove that if \(W \ne D\) and \(OW = OD,\) then \(DZ\) is tangent to the circle \(AXY.\)