12.1.9 Olympic geometry

Problem: On the side \( A C \) of the triangle \( A B C \) the point \( D \) is chosen. Circles, inscribed in triangles \( A B D \) and \( C B D \), are tangent to the segment \( B D \) at points \( P \) and \( Q \). What is the minimum natural value of \( x \) for which we can determine the relation \( A D: D C \) one-valued, if it is given that \( A B=21, B C=27, A C=30, P Q=x \) ?