How would you react if something may not be possible according to physics but somebody may furnish a mathematical proof that may falsify it, as has happened in the case of the concept of barycenters? 

As explained by NASA in its post https://spaceplace.nasa.gov/barycenter/en/ taking the example of a seesaw using the following diagram, barycenters are imaginary points about which it should be possible for a star to balance its planets in the same way as two persons swinging on a seesaw may balance each other if the values of the torques exerted by them at the fulcrum of the seesaw may be same.    

Though according to this post, the barycenter of a star and its planet is supposed to behave just like the fulcrum of a seesaw, you would be surprised — such a view does not go well with the basic principles of physics.

Therefore, it is wrong to compare the manner in which a star and its planets balance each other with the manner in which the persons sitting on a seesaw balance each other.

Because  the stars and their planets don’t rest on any physical object of the type of the plank over which two persons sit over a seesaw nor any such plank rests on any physical fulcrum.

Actually barycenter of a star and its planets is the point where the gravitational force exerted by the star and the gravitational force exerted by its planets balance each other.

I don’t think it needs such a complicated mathematical explanation as has been given in the following paper. https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-29/issue-4/Barycenters-and-extreme-points/10.2969/jmsj/i02940607.pdf ^[1]

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1. Math. Soc. Japan, Vol. 29, No. 4, 1977