Is it a thereom or an axiom that two planes intersect only in lines?

Question

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?

Answer 1

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,