>Do you have a nice one for the circumcentres?

Perpendicular bisectors of the bases move with equal velocities. For each of the triangles ABD and ACD, this movement causes the circumcenters to glide along the perpendicular bisectors of AB and AD, which do not move. These motions, too, have fixed velocities. Their components parallel to the base are equal. Only these components are responsible to the point of intersection of the lines through the circumcenters with the base. Thus the two lines intersect on the base for all positions of C, if they do for one. This one can be taken as the foot of the altitude from A, in which case they meet in the midpoint of BD.