I was looking at this web page
http://www.cut-the-knot.com/ctk/CompleteQuadrilateral.shtml
and it states, near the bottom:

For the centroid, the circumcenter and the orthocenter it is quite clear that if the lines intersect on BD for one position of C, then the same is true for all other positions as well. (What is lost or gained on the left from the common point of intersection, is gained or lost on its right.)

I've been thinking about it, but I don't see this at all. Can anyone help?