Symmetric nested octagons

Consider a symmetric octagon ABCDEFGH. Take its side-middles and construct a second octagon with vertices at these points. Repeat the above construction several times, creating a sequence of polygons P(1), P(2), ... etc. Show the following properties:
1) All these polygons P(n) are symmetric about the symmetry center Y of the original polygon.
Show also that for growing n the following properties are true.
2) The lines joining the middles of opposite sides of a polygon P(n), and P(n+2) tend to a fixed line.
3) Directions of corresponding sides of P(n) and P(n+2) tend to a fixed one.
4) Lines joining corresponding vertices of P(n) and P(n+2) tend to pass through Y.
5) The ratio of corresponding sides of P(n) to P(n+2) tend to a fixed number.
6) The ratio of the perimeters of P(n) to P(n+2) tends to a fixed number.
5) The ratio of the perimeters of P(n) to P(n+1) tends to a fixed number.
Give generalizations for arbitrary symmetric equilateral 2k-gons. Look however at SymmetricNestedHexagons.html for a particular interesting case, where the above defined sequences of numbers are constant.

What happens with non-symmetric equilateral polygons? More generally, one could consider arbitrary polygons, not necessarily equilateral and ask the same question. One could even create sequences of polygons, taking not the middles, but points at a fixed ratio k from the extremities of each side. Look at Nested_Polygons.html , to see what I mean.
Concerning the construction of the nested polygons, notice that the scheme (b) has been plugged into the main scheme several times. This has be done using the [Scheme_Sockets] tool.

Variable points are: A and B, using the selection tool (Ctrl+1), and C, D, E using the Select-on-contour tool (Ctrl+2). The three last points move on hidden circles.