I know that two planes always intersect in a line or are parallel to each other, and there's an question about two planes in a point: Can two planes intersect in a point?. But how can you prove that two planes can't intersect in any other shape, like a parabola, a circle, a rectangle or basically any function other than an line? Or is it an axiom that we can't prove?
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It depends on how you set things up. – Qiaochu Yuan Dec 08 '20 at 01:20
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Planes are solutions of linear equations. Any line (which must lie in both) must also be linear... hence a line. QED – David G. Stork Dec 08 '20 at 01:26
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1Also, planes are convex sets, so the intersection has to be convex, which eliminates some of the possibilities you've raised. – Dave L. Renfro Dec 08 '20 at 15:44
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As said in the comments there is a proof by analytic geometry: Any plane is an solution to a linear equation, as such: $$ax+by+cz=d$$ $$ex+fy+gz=h$$
Any line which lies in both planes thus must also be linear, so that's why it's a line.
However, I do not think it is possible to prove that in only Euclidean geometry,