Ok, thank you, I see that now. Basically, the two lines through C and C itself move in tandem. I was getting confused becuase sometimes, when I clicked on the java applet to move the point C, point A would move as well, which completely obscures the argument.

I see the argument for the centroids, that's pretty clear. I see how the argument goes for the orthocentres as well. Do you have a nice one for the circumcentres? (Other than appealing to the Euler lines of triangles ABC and ACD.)