Question 72
Let $ABC$ be a triangle and $D$ a point on the side $BC$. The tangent line to the circumcircle of the triangle $ABD$ at the point $D$ intersects the side $AC$ at $E$. The tangent line to the circumcircle of the triangle $ACD$ at the point $D$ intersects the side $AB$ at $F$. Prove that the point $A$ and the circumcenters of the triangles $ABC$ and $DEF$ are collinear.