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Written by VVilliam :coolS: on 2024-09-14 at 06:25

Does any line through the barycenter of a triangle cute it into two polygons with the same area?

Random thoughts at 3 AM.

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Written by VVilliam :coolS: on 2024-09-14 at 06:29

Random thoughts at 3 AM turn out to be stupid, disregard.

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Written by VVilliam :coolS: on 2024-09-14 at 08:41

Wait no it actually works.

Physicist's argument (can be converted into proper argument using integral definition of centroid):

Place your finger ("any line") under a paper triangle such that the barycenter is supported. Since the barycenter is the center of mass, the piece of paper will stay balanced. Since it's not tipping over to one side or the other of your finger, both sides must have equal mass, thus (assuming uniform density for the paper) equal area.

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Written by VVilliam :coolS: on 2024-09-14 at 09:02

Now that I think about this, doesn't this apply to all shapes, not just triangles?

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Written by Joseph Hertzlinger on 2024-09-15 at 17:36

@wqferr

I responded but deleted. Is this a homework problem?

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Written by VVilliam :coolS: on 2024-09-15 at 22:23

@jhertzli Nope, genuinely a random thought at 3 AM. I've responded to this with a solution myself, though I'm not entirely sure it holds water.

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