Makman is right!!!
Not absolutely, but right.
If you keep speed and course through all the process this is what happens:
The construction of a possible solution course (among the infinite existent) implies choosing a point along the second bearing. What if you choose your own second position as a starting point? you will end up with your own fourth position as the protracted one for your target. Therefore, the red line always goes through your fourth position. So if you don't change speed or course along the process, your fourth bearing will be meaningless. In the document that explains the method, I had considered meaningless the submarine position, so I put on some arbitrary ones and all went well, because it seems that slight variations on speed or course result in wide variations in the red line, under feasible circumstances, which is enough to provide an accurate solution.
I was quite perplex when I realized that, after Karamazov's statement that he had not changed course. After reviewing his snapshot, I notice that he had at least changed speed between his first and third bearing. That's enough. I have yet to assess to what amount the direction vector of the red line is sensible to those changes, but I can assure you that you need at least one change in speed or course anywhere along the construction to make your fourth bearing meaningful, (not necessarily right before the fourth bearing, and that's the only reason I said not absolutely right at the beginning, but he is essentially right: it is not that your bearing will coincide with the red line, but that it will cut it through your own position, a point that we mathematicians call a singular point -and we sailors call collision point- where information has vanished. That is, roughly: the bearing to a ship in your own position is any).
I will rewrite the document posted in this thread to reflect those facts and post it here.
Thank you, Makman. Well seen. This is collaboration at its best.
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