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There are at least two questions on this site related to the (somewhat famous) false proof that all triangles are isosceles as presented for example in M. J. Greenberg's 'Euclidean and Non-Euclidean Geometry'. The fallacy itself can be read on this site here:

What is the flaw in this proof that all triangles are isosceles?

Although the answers for both of these questions give the correct reason that there is an incorrect assumption of 'betweenness' leading to the fallacy, I have not seen an attempt to prove the betweenness part of the fallacy itself, which is the crux of the matter (excuse me if I overlooked something in the other two answers).

I struggled to come up with a proof for the betweenness part of the fallacy. I had added an early version of this effort as an answer to one of the other two questions earlier, but did not get any feedback. I think this betweenness part deserves a separate question: Can anyone critique the basic steps of my proof as given here and/or give a proof for the betweenness part of the fallacy?

The diagram covers 'case 4' of the false proof, that the intersection of the perpendicular bisector and the angle bisector are at a point D exterior to the triangle. The other false case of D interior is similar, which I describe in the diagram text. I have left the last case, that D is on the triangle (even though the perpendicular bisector and angle bisector do not coincide) aside for now, but if the argument I give here is correct, I think it too will be an extension of the same idea.

enter image description here

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    What I'm saying is, suppose you have a triangle such that the perpendicular bisector and angle bisector intersected inside the triangle. Then, following the correct subsequent logic of the false proof the triangle is iscoseles. Therefore, if it is not isosceles, the angle bisector and perpendicular bisector CANNOT intersect inside the triangle – aman Mar 24 '20 at 16:46
  • @aman I am not able to understand how you can base a proof on 'correct subsequent logic of the false proof', it's an oxymoron. –  Mar 24 '20 at 16:53
  • The false assumption of the proof is used (with correct subsequent logic) to imply that "all triangles are isosceles" Thus, if a triangle is not isosceles, this false assumption cannot hold – aman Mar 24 '20 at 16:57
  • I know a lovely proof-by-contradiction that in all cases when the two sides of the triangle meeting at $C$ are not equal, then either ($(AFB)$ and $(EBC)$) or ($(ABF)$ and $(BEC)$). For if you assume otherwise, you can prove that the triangle is isoceles! ;-) – Paul Sinclair Mar 25 '20 at 02:48
  • @aman I think I see what you are saying, but you are assuming the conclusion. Sorry if I'm misunderstanding you. The other two questions refered to the issue that the false proof mistake is an assumption about betweenness. I think a proof of that is req'd which is what I am trying to do. The structure of my proof is basically, suppose there are some 'weird' isosceles triangles where the angle bisector and perp bisector intersect interior/exterior. Then both of these cases reduce to the 'normal' case, because of betweenness. –  Mar 25 '20 at 15:19
  • @PaulSinclair Precisely! A version of that is what I have attempted (see prev comment) -- but I have not been able to find it written down to see if my development of the idea is correct. If you know a reference to someone else's version, would like to know. –  Mar 25 '20 at 15:24
  • @Circulwyrd - my comment was actually a tongue-in-cheek (or maybe just "cheeky") way of pointing out that the "false proof" is effectively a proof-by-contradiction of the betweenness condition. The only difference is that the proof-by-contradiction recognizes that the point ordering is an assumption, which is then contradicted by the false result, where as the false proof pretends no assumption was made. (This is not intended to say you shouldn't bother with a more-direct proof. I was just pointing out that it does amount to a proof itself.) – Paul Sinclair Mar 25 '20 at 16:38
  • @PaulSinclair I think you and aman make the same basic argument: That the end triangle is not isosceles is enough for a contradiction. I suppose ... A quote from Coxeter on an ordered geometry section: "It may seem to some readers that we have been using self-evident axioms to prove trivial results. Any such feeling of irritation is likely to evaporate when it is pointed out that the machinery so far developed is sufficiently powerful to [solve] Sylvester's conjecture which baffled the world's mathematicians for forty years." In other words, to delve into the betweenness is a good thing. –  Mar 26 '20 at 14:34

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