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that the y coordinate of the center of mass must lie on this line,
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because if I take y to - y,
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it maps onto itself so it looks the same.
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But it's supposed to reverse Y;
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therefore, Y is -Y and therefore it's 0;
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so, it's evidently lying somewhere on this line.
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I cannot do further calculations of this type
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by saying where it is on the line,
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because it has no longer a symmetry in the x direction;
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it's symmetric on the y after flipping y,
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but I cannot take x to - x.
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In fact, if I take x to - x