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In science, there are no proofs, per se. You can only disprove.

Generally, when speaking of the center of the galaxy, one refers to the center of the galactic mass. A galaxy revolves around its center of mass, and we can see a galactic bulge in its center.

We know this from observations of other galaxies, and Doppler images of its motion at the edges, when seen edge-on, one side will be slightly red-shifted relative to the other.

In our galaxy, by mapping the known stars in a three-dimensional grid, we can see a central bulge and spiral arms, similar to what is seen in many other galaxies. This is sufficient to know that we are in an arm of the galaxy, about 2/3 of the way from the center to the rim.

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