Let's change the circles while keeping them tangent and inscribed into the angles of parallelogram ABCD. In the applet below me can do that by dragging the center of one of the circle.

The centers move along the angle bisectors of the corresponding angles and the center line remains parallel to itself. This feature implies that the line preserves its length: assuming the two circles are tangent to each other and inscribed into two opposite angles of a parallelogram, the direction and the length of the line joining the centers does not depend on, say, the location of the point of tangency on the diagonal.