What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon? Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice.

Note that removing the "points" of the star yields a similar heptagon. I want to know the side length of this internal heptagon (blue sides) in relation to the side length of the original heptagon. 
The blue lines: $$ \rho = 2cos(\pi/7)
$$ 
The green lines: $$ \sigma=4cos^2(\pi/7) - 1 
$$
[Golden Fields: A Case for the Heptagon, Peter Steinbach, Albuquerque Technical-Vocational Institute, Mathematics Magazine, Vol. 70, No. 1, February 1997]
No idea where to start :(

Refer to the following diagrams:-

