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May 8 at 16:53 comment added Nandakumar R Thank you again. And I think the nice thing about this argument is its independence from whether the plane is euclidean or otherwise.
May 8 at 7:29 comment added Fedor Petrov I hope that the answer covers both parts
May 8 at 7:29 history edited Fedor Petrov CC BY-SA 4.0
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May 8 at 7:01 history edited Fedor Petrov CC BY-SA 4.0
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May 8 at 3:49 comment added Fedor Petrov "only if" part is rather clear. For a symmetric regions bisectors pass through a center of symmetry
May 8 at 3:42 comment added Nandakumar R Thanks. As far as I can make out, this argument also settles the claims in the euclidean case as well and can replace the two proofs at mathoverflow.net/questions/448712/…. What does the phrase "if part is rather clear" mean?
May 8 at 3:39 vote accept Nandakumar R
May 7 at 21:29 history answered Fedor Petrov CC BY-SA 4.0