As in Ring of triangles I, ABCDEF is a hexagon and O is an interior point. Triangles ABO, CDO, EFO are is equilateral. Their respective side lengths are different integers with no common divisor. (They are also not 4,3,7 because that was part I.)

BC, DE, and FA are also integer lengths.

Find a set of lengths {AB,BC,CD,DE,EF,FA} for which this is possible.

Note: There are sides of the hexagon that can be the same.