Must all lines of symmetry for a shape pass through the same point?

Question

Must all lines of symmetry for a shape pass through the same point?

I'm unable to think of a shape that doesn't follow this rule but can't come up with rigorous proof for this.

By shape I mean any sort of closed figure with finite area.

I'm new to this site and don't know much about formal math lingo, please forgive any of my mistakes.

Any help would be appreciated

Answer 1

Yes. Every line of symmetry has to pass through the center of mass / centroid, because by definition the mass on both sides of a line of symmetry has to be the same.

(Since the other answer has raised the possibility that the shape could be $\mathbb{R}^2$ let's stipulate that the shape should be bounded by something like a piecewise smooth curve.)

Answer 2

I'm not sure about what you call a 'shape'. But for $\Bbb R^2$, every line is a symmetry line, and obviously they don't meet at a single point.