[alogo] Symmetric hexagons sequence

Start with a symmetric hexagon ABCDEF and consider its inscribed conic. The touch points with it build another symmetric hexagon GHIJKL. The construction can be repeated in both directions, inwards and outwards creating a sequence of symmetric hexagons extending also in both directions.
Some interesting infinite sets are created:
[1] An infinite set of conics (in the case of an ellipse, is it convergent inwards, where?).
[2] An infinite set of symmetric hexagons (same question).
[3] An infinite set of vertices of the hexagons (find the paths on which they are arranged).
[4] An infinite set of foci of the conics (find the curve on which they are arranged).

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

The polar cX of any point X (from the infinite set of vertices of all hexagons) with respect to any conic (out of the infinite family again) is a line passing through three other vertices.
The whole configuration can be considered to start from a triangle LGH, formed by two consecutive sides of some hexagon and point P, the center of all hexagons. Take then the symmetric of the vertices of the triangle w.r. to P, defining the first (foundamental) hexagon, and repeat the whole construction. When does it happens that the analogous construction from another triangle, like G'H'L', gives the same infinite sets?

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