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There is following statement: if a square is cut into 2 congruent parts (i.e. equal area and shape), then the cut line (not necessarily straight) will have point reflection around the center of the square, in particular, it will pass through it. I think it's right, but I cannot prove it. Maybe anyone has any idea?

Thank you in advance.

user3798254
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    It would be better if you showed some steps. for example why you think it's right, any attempts at a proof, etc. – KingLogic May 10 '21 at 19:44
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    To add to the above comment of @KingLogic -- can you say why the cut line must pass through the center? [There could conceivably be such cut lines with point reflection around the center but not passing through it.] – coffeemath May 10 '21 at 19:49
  • I cannot show any steps because I don't know, how to prove it. All the examples for cutting a square into 2 congruent parts, that I know, have the symmetric cut line. But the question is: how to prove it? – user3798254 May 10 '21 at 19:58
  • I think you could prove this through contrapositive, ie, show that if a cut line does not have point reflection about the center then it cannot cut the square into two congruent pieces. – Jake Brown May 10 '21 at 22:04
  • Thank you Jake, I'd also prefer a contrapositive way, but it needs more details. Actually, I don't think there is a simple proof, because any simple proof would probably consist of the same words for a mirror symmetry. But there are cut lines without mirror symmetry. – user3798254 May 11 '21 at 01:09

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