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The barycenter (or barycentre; from the Greek βαρύκεντρον) is the point between two objects where they balance each other. For example, it is the center of mass where two or more planets orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting around a point that lies outside the center of the larger body. For example, the moon does not orbit the exact center of the Earth. It actually orbits a point on a line between the center of the Earth, and the Moon. This is about 1,710 km below the surface of the Earth. This is the point where the mass of the moon, and the mass of the Earth balance. This is the point about which the Earth, and Moon orbit as they travel around the Sun.
The solar system of course also has a barycenter—depending on the current locations of the planets the barycenter of the solar system is below the surface of the sun or more than twice the sun’s diameter outside of the sun's surface.
Also the barycenter exists in the world of geometry. For example, the circle has a barycenter.