Question 136
A point $D$ is chosen on the side $BC$ of an acute triangle $ABC$ (and is distinct from the vertices). Points $P$ and $Q$ are defined as the centers of the circumcircles of the triangles $ABD$ and $ACD$ respectively. Prove that, as soon as the triangle $ABC$ is fixed, there exists a point in the plane, which is different from $A$ and belongs to the circumcircles of all the possible triangles $APQ$ (generated by the different positions of the point $D$).