City problem · Geometry

Let \(ABC\) be a triangle. The tangents to the circumcircle of \(ABC\) at the points \(B\) and \(C\) intersect at a point \(X\). The circle through \(A\), \(B\), \(X\) meets the line \(BC\) again at a point \(P\), and the circle through \(A\), \(C\), \(X\) meets the line \(BC\) again at a point \(Q\). Prove that the tangents to the circumcircle of triangle \(XPQ\) at \(P\) and at \(Q\) intersect at a point of the line \(AX\).

The configuration: tangents to the circumcircle at \(B\) and \(C\) meet at \(X\); the circles \(ABX\) and \(ACX\) (dashed) meet line \(BC\) again at \(P\) and \(Q\); the tangents at \(P\) and \(Q\) to the circle \(XPQ\) meet at \(Y\), which is claimed to lie on the line \(AX\).