Are there always $2(n-1)$ ways to intersect two polygons such that these intersections are convex polygons with $3,...,2n$ vertices each?

Question

Suppose two convex polygons with $n$ vertices each.

Proposition: There are always $2(n-1)$ ways to intersect them such that these intersections are convex polygons with $3,...,2n$ vertices each.

Is this proposition true?

Is there any theorem related to this?

I found this "fact" drawing something like this:

For $n=4$

Which other conditions I have to consider for this? Like size for example.

Thanks for your help.

Answer 1

You cannot intersect these two squares to obtain an $8$-gon. The best you can achieve is a $6$-gon (arranging the small square near a corner of the large square).

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So indeed size must be taken into account.