My point was that you cannot establish a “half-way” using only two positions. You need more.
Yes, but then the point that’s equidistant from either end is not the line itself. The line itself has no middle until you impose or construct one on it using other, stipulated positions - which, if imposed, would change the scenario that was presented.
How do you know that the middle of your line shows the middle of a line?
A line? What line? My line’s a line very similar to the line you were asking about a while ago. Any line will do as long as it has limits and the middle can be indicated as an area equidistant from the ends
Why does it need to be? ‘Point X is the middle of the line’ doesn’t entail ‘Point X is the line’.
JJ,
What is the point of this thread?
Dude, where’s my car?
In order to establish a middle that is “equidistant” I need to establish a middle. How do I do that?
You can’t establish a half-way position using just the two points. You have to make a new line, for example by drawing a line at right angles etc etc.
if someone says, “I’m in the middle of a thought”, what does that mean?
Yes you can. You measure the distance between the two points, divide it by two, trace that calculated distance from either of the two points, and make the place where you end up your third point.
Your third point might not be ‘the line itself’, but you haven’t yet explained why it has to be for it to be ‘the middle of the line’.
Define middle and two points.
Conceptually yes actually probably unless you are talking of non euclidean forms of geometry then they can meet in the ends or anywhere depending how you define your axioms or maths.
You can in 2 dimensions. 
Triangulation is the word you are looking for though.
Points at the end of a line are not positioned relative to the line except at the end of a line. so I can’t divide or subtract and divide the positions by two. I think you are placing this line in a space where positions can be assessed. That would constitute a new construction.
I’m happy with your, standard, definition.
Good. mathematically its sound at least. 
I didn’t say anything about dividing the positions by two. What I said is that you divide the distance between the two points by two.
You’ll have to say how you are going to divide the distance without being given the positions, and the sort of construction on offer that assesses the two half’s as “equal”.
Sorry, I don’t understand what you’re getting at. The two positions you start with are the positions of the two ends of the line. You gave me those when you drew the line!
Again, I don’t understand what you’re getting at. If you divide the length of the line (i.e. the distance between the two ends) by two, you’ll have two equal halves. That’s basic arithmetic.
Yes, I gave two points at the end of the line. But without knowing their “positions”, that is, without knowing their coordinates in a spatial framework, I can’t see how you can find the middle of the line.
What I was alluding to was that the definition of “half-way” or “half” can be no more or less than a function of the construction you used to create or show what is called “half-way” or “half”. But then that is a conceptual difficulty of arguing for an absolute position, rather than a difficulty in construction.
Why can’t the two points themselves provide the framework, so that the middle is specified only relative to the two points?
No problem at all then.