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I was reading Euclid's Elements E-book I found online and got stuck on this concept. I will just copy what I found to be very absurd.

There could still be another different triangle with the same sides. [For instance there are usually two different triangles with the same SSA data, but they are not close to each other, so you cannot wobble one into the other. So showing that a triangle cannot wobble when certain measurements are fixed only suggest there are only a finite number of such triangles, it does not really argue there is only one.]

I don't really understand what he's saying here... What does it mean when triangles wobble? I just can't picture another different triangle also has same sides.

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  • $\begingroup$ Could you provide a link to the book please? $\endgroup$
    – Asinomás
    Jul 26, 2016 at 18:07
  • $\begingroup$ Found it, here is a link: alpha.math.uga.edu/~roy/camp2011/10.pdf $\endgroup$
    – Asinomás
    Jul 26, 2016 at 18:08
  • $\begingroup$ According to the OP's title, it concerns an SSS case, at least that is how I interpret it... $\endgroup$
    – imranfat
    Jul 26, 2016 at 18:09
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    $\begingroup$ If three sides are given, then the triangle is indeed determined. The author is presumably arguing that this requires proof, and in particular that an informal rigidity argument is not sufficient, because SSA is insufficient. $\endgroup$
    – André Nicolas
    Jul 26, 2016 at 18:19

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I think what the author means is as follows:

If you build an actual triangle using rigid straws and you try to move one of the vertices, then the whole triangle will move rigidly (so it won't be deformed). So a triangle with sides $A,B,C$ cannot be "wobbled" into a different triangle with sides $A,B,C$.

The author seems to refer to this experiment as a "flawed proof" for the uniqueness of a triangle with given sides.

His argument is: Maybe there are exactly $2$ triangles with sides $A,B,C$. If this was the case then we would not be able to wobble one of them into the other, because to go from triangle $1$ to triangle $2$ we would have to pass through an infinite spectrum of different triangles, which would not have lengths $A,B,C$. (And this is impossible with our "straw" version of the triangle, since the sides must stay of the same length).

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