Edit: this answer is not correct. It solves for undirected cycles (and I believe does so correctly) but the number for chains is different. I will not be solving for chains. Answer left in place in case it is somehow useful for future attempts.
This is pretty basic math, if a bit convoluted. There are 6 non-intersecting polygonal chains that cover all 7 points (depending on which direction you point the slice). Anything else would intersect.
Further, the chords that cross the center are non-legal, as they'd cause the center point to be touched more than once.
The total number of polygonal chains if you ignore those limitations? Simply 6!/2. You start at A. You have 6 options of where to go. After that 5, after that 4, and so on. Then you cut the result in half to acknowledge that it's not a directed graph.
The number of graphs where a line cross over the midpoint is equivalent to the number where B and E is one point, plus where C and F are one point, plus where A and D are one point, less the number where more than one of those is true. In this case, that means 3x5!/2 - x. Th x is the cases where two of those are true (subtract once, because it counts once rather than twice), less the case where all three are true (which shows up 3 times in the minuend, but three times also in the subtrahend, so we need to add it back in because it's supposed to show up once).
So, the total number of cases where it crosses the midpoint at least once without stopping is (3x5! - 3x4! + 3!)/2. The non-intersecting is, as mentioned, 6.
The final result is (6! - 3x5! + 3x4! - 3!)/2 - 6, which (I believe) comes to 207