A Geometry Puzzle in a Regular Nonagon

Two chords are drawn on a regular nonagon from the midpoint of a side to vertices that are three and four vertices away and three edges apart. A line through the midpoints of these chords is parallel to a diagonal of the nonagon and close to it. What is the distance between that line and that diagonal? The regular nonagon has side length one.

Point M is the midpoint of edge AB of a regular nonagon. Two chords MD and MI are drawn from M to vertices D and I respectively. A line is drawn through the midpoint of these chords. It is parallel to the diagonal DI of the nonagon and close to it, but just a little further from the centre of the nonagon.
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