In the acute-angled triangle \(ABC\), the point \(F\) is the foot of the altitude from \(A\), and \(P\) is a point on the segment \(AF\). The lines through \(P\) parallel to \(AC\) and \(AB\) meet \(BC\) at \(D\) and \(E\), respectively. Points \(X \ne A\) and \(Y \ne A\) lie on the circles \(ABD\) and \(ACE\), respectively, such that \(DA = DX\) and \(EA = EY\).

Prove that \(B, C, X,\) and \(Y\) are concyclic.