A nice conjecture about the centroid of a triangle.

ABC is any triangle. Points D, E, F are the vertices of similar triangles constructed upon the three sides of the triangle. You can change the shape of the similar triangle by moving point D. The conjecture is that the centroid of triangle ABC is the same as the centroid of triangle D,E,F.
  • Can you prove the theorem?
  • Can you make further generalizations?