What is the proof the centers of mass lie along lines of symmetry and vise versa?

Question

I was reading this and notices multiple references to the fact that lines of symmetry and centera of mass are deeply interconnected, and I was wondering if anyone could give me a proof, or an intuitive explanation on why this relationship is true.

Answer 1

Here's an intuition:

The idea of a line of symmetry is that you can flip the object over (keeping that line in the same place) and the flipped object will be identical to the one you started with. Any feature of the object that is not on the line of symmetry therefore matches an equal feature on the other side of the line. When you flip the object over, you swap the two features. You can't have the feature in just one place, because then when you flipped the object over, it would be different from the original object--the feature would have moved.

But as you can see from the formulas for center of mass, there is exactly one center of mass of an object. There will never be two or more centers of mass. So the center of mass must be on a line of symmetry; if it were anywhere else, the symmetry of the object would force it to have two centers of mass.

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The paragraphs above approach the conclusion in a somewhat round-about way for the sake of rigor: there's a proof by contradiction hidden in there. More succinctly:

The mirror image of the object over its line of symmetry is identical to the original object, including the location of the center of mass. Therefore the center of mass must be at one of the points that does not move when it is mirror-imaged.