I've had this question in my head recently due to my math teacher giving me this problem as a bonus on a test.
So I have two regular hexagons inscribed in two distinct circles, of radius n. The two circles intersect, the intersection points being vertices of the hexagons. Let them be O1 and O2.
Part (a) of the question was to prove the existence of a circle of radius n that passes through 2 vertices of the first hexagon and 2 vertices of the second, while containing only ONE of the intersection points.
So suppose that this circle intersects one hexagon at vertices A and B and the other at vertices C and D. This circle, while intersecting at A, B, C, and D, would also have to include one of O1, O2, in its interior.
Part (b) of the question was to prove/disprove that these 4 points (A, B, C, D) can make up the vertices of a regular octagon/decagon/dodecagon.
Nobody in my class, including myself, got the bonus.
I've thought about this question for a while, and I've given up.
My teacher also proposed the same problem, but with pentagons and heptagons, and to prove/disprove existence of hexagon/octagon/decagon for pentagon, and decagon/dodecahedron/14-gon for the heptagon.