For what it’s worth, there are animations on Wikipedia showing how barycenters, the common point about which two bodies rotate, work. Look here:
en.wikipedia.org/wiki/Center_of_mass Links I post don’t seem to work, so you might have to copy/pase, or type.
And also fyi, you can take any body, say the moon or an asss-teroid such as a geocentric mind, and do math that “proves” it to be the center of whatever. All that’s happened is that given an assumption, its consequent parameters can be plotted. So if you want the Earth to be the center of things, fine, there is the math that describes how that works. But it doesn’t make that assumption true. The math is neutral and bends to the application required by the asss-umption, in this case geocentrism.