anchor points at intersecting paths using exclude overlapping shapes

Hello,

I am trying to make a segmented circle using shapes. Every time I use 'exclude overlapping shapes' functionality, I always get different results. I'm not sure on what causes this but my objective is to be able to have those extra anchor points on interesting paths (to where the arrows are pointing). How do I do this? Am I missing extra steps? 
(I can only access PS at the moment 😞 )

thank you so much!