Chop polygon boundary into segments based on points

I'm using a geometry coercer to coerce a polygon to a line followed by a point on area overlayer to chop  the line/boundary into segments.

 

 

In some cases the start/end point of the original polygon leads to two segments/line features being created either side of the original start/end point where I only want to have one. I can compare the resulting segments to the original start/end point choose to join the two together to get the correct number of segments but in some cases the start/end point aligns with a point I am chopping at, i.e. features either side of the start/end point should remain as separate line features.

 

 

Any ideas?