Let \(ABCD\) be a cyclic quadrilateral. Assume that the points \(Q, A, B, P\) are collinear in this order, in such a way that the line \(AC\) is tangent to the circle \(ADQ\), and the line \(BD\) is tangent to the circle \(BCP\). Let \(M\) and \(N\) be the midpoints of segments \(BC\) and \(AD\), respectively. Prove that the following three lines are concurrent: line \(CD\), the tangent of circle \(ANQ\) at point \(A\), and the tangent to circle \(BMP\) at point \(B\).