How many common points do two regular polygons in a circle have?

Question

I was attempting to solve the following question:

In a circle you have a $27$ sided regular polygon and a $297$ sided polygon $($all vertices are on the circle$).$ How many common points do they share?

I thought that $n\choose r$ would be helpful but I don't have any idea about how to use it.

Answer 1

HINT: Both polygons consist of straight line segments (the edges), and both are convex. Hence each edge of the one polygon intersects at most two edges of the other polygon. (Prove this!)

Answer 2

I observed that a line of 27-gon have 11 sides of 297-gon in minor segment if we consider 1point of both polygons coincide.

Also when 1point coincide than each point of 27-gon coincide by simmatry

So only 27 points are common

If I consider no point coincide then 1line of 27-gon intersect with 2 sides of 297- gon so there are 54 points common.