Why are you trying to complicate things?
1. Just use the Pen Tool and draw the 3 shapes. If you worried about the lines at the mitre join just go to Outline (ctrl+Y) and then you will be able to make sure that the line converge on top of each other.
2. If you are working from a full shape remember you have the Scissor Tool and you could break apart at the join and then use the Pen Tool to join up the broken areas.

To use Illustrator properly you must learn to use the Pen Tool it is a must. Sorry if I have not answered your question but it seems to me so simple.