In astronomy, a barycenter is the center of mass of two or more celestial bodies that orbit each other, or the point at which the objects are balanced. When an object is traditionally thought of as orbiting another, such as the Moon orbiting the Earth or the Earth orbiting the Sun, the center of orbit is virtually never at the direct center of the more massive body. Rather, both objects are orbiting the same point, the barycenter, that may lie more or less off center within the more massive body.
The larger the difference in mass between two objects in the same orbit system, the larger the discrepancy in size between their respective orbits. Two objects of equal mass orbiting the same point may have travel in the same orbit and lie at opposite points on it, or they may travel in divergent elliptical orbits around the barycenter. In a system like that of the Earth and the Sun, on the other hand, the more massive body hardly moves at all compared to the less massive one.
The barycenter of two bodies orbiting each other can be found by calculating the distance between the more massive body and the barycenter. This distance is given by the distance between the centers of the two bodies, multiplied by the mass of the smaller body divided by the combined mass of both bodies. If r_{1} is used to represent the distance between the center of the larger mass and the barycenter, a is used to represent the distance from the center of one mass to the center of the other, and m_{1} and m_{2} are the masses of the larger and smaller bodies, respectively, the following equation can be used: r_{1} = a * (m_{2} / (m_{1} + m_{2})). If r_{1} is less than one, the barycenter is located inside the more massive body.
anon278213 Post 2 
In general (in 3D) they do not. If they have the same shape and are homogeneous (e.g. two planets or spheres) etc and oriented the same way, then yes. What you need to do conceptually is slice each object into very tiny parts; a slice of the same size in the core of the earth contributes much more than a slice of the same dimensions in say, the ocean. 


anon42257 Post 1 
please next time tell us about why if two objects have the same weight their barycenter are exactly half way. why not if the other object is bigger? 