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I know this is a simple question but I cant find the name for this.

(For a figure go to the bottom of the question)

I know that for similar triangles $\Delta ABC$ $\Delta A'B'C'$ I can divide any two similar sides and get the scale factor $K$

$\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{AC}{A'C'}=K$

The scale, of course, changes for different similar triangles. However, if I where to do some algebra.

$\frac{AB}{A'B'}=\frac{AC}{A'C'}$

$AB=\frac{AC}{A'C'}A'B'$

$AB=\frac{A'B'}{A'C'}AC$

$\frac{AB}{AC}=\frac{A'B'}{A'C'}=m$

I get to some constant $m$ that does not change between any similar triangles.

I also noticed that has some kind of relationship with the law of sines. I want to know:

1)what is the name of this constant or how can I search for it (could not find it).

2)If there is an analogus euclid element's proposition about it.

Here is an example on geogebra. If you drag the "drag me" point you are generating a diferent triangle $A'B'C'$ similar to $ABC$ and you can see how $K$ changes but $m$ stays the same for both triangles independently of their difference in size https://www.geogebra.org/classic/kbwzynd6

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It doesn't have a name, unless the triangle is right-angled.

You probably noticed that there are six such constants: your $m = \frac{AB}{AC} = \frac{A'B'}{A'C'}$, its brother $n = \frac{BC}{BA} = \frac{B'C'}{B'A'}$ and its sister $p = \frac{CA}{CB} = \frac{C'A'}{C'B'}$ together with their inverses.

All six are the ratio between two sides meeting in a point and you don't have any more information than that to tell them apart. If I give you a random triangle (but do not tell you the name of the vertices) you have no way to say which one is $m$ which one is $m^{-1}$, which one is $n$ which one is $n^{-1}$, which one is $p$ and which one is $p^{-1}$.

That is not a very fortunate situation. The reason that these constants don't have a name is that if they had and I were to use that name you still wouldn't know what constant I was talking about.

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    $\begingroup$ indeed I noticed. That is very interesting. I have seen them used in proofs about conic sections extensively by the ancient greeks and just recently noticed that they where not talking about the "similarity ratio". It's sad there is no name for them, apollonius uses them quite a bit. $\endgroup$ – Joaquin Brandan Aug 26 at 19:20
  • $\begingroup$ You mentioned "unless the triangle is right-angles", what did you meant by that? $\endgroup$ – Joaquin Brandan Aug 26 at 19:50
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    $\begingroup$ Well suppose that we give a name to the size of angles of the triangles, e.g. write $\alpha$ for the angle under which AB and AC meet. Then your $m$ is either $\sin(\alpha)$ or $\csc(\alpha)$ and unlike in my answer it is perfectly well defined which of the two $\endgroup$ – Vincent Aug 26 at 20:06
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    $\begingroup$ So suppose that we are facing a situation where $m$ equals $\sin(\alpha)$ then we can say that $\sin(\alpha)$ is the name of the ratio. Of course this still depends on assigning the name $\alpha$ to the angle first $\endgroup$ – Vincent Aug 26 at 20:08
  • $\begingroup$ awesome. That clears some things. I happen to have a bunch of these ratios in non rectangular triangles in old greek texts (so trig functions where not a thing). Do you by any chance know about any reference I could read about them? $\endgroup$ – Joaquin Brandan Aug 26 at 21:25
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The constant $K$ is often called "similarity ratio" or "scale factor". The constant $m$ does not have a definite name (there are various constants of this type). As regards Euclid's propositions, the most important in this context are those from Book VI, from VI-4 to VI-7.

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  • $\begingroup$ Isnt scale factor the name for $K$ ? $\endgroup$ – Joaquin Brandan Aug 26 at 19:06
  • $\begingroup$ I have reviewed the propositions from euclid and the names "similarity ratio" and "scale factor" and it would seem you are answering the question "what is $K$" and not "what is $m$". Am i wrong? maybe I'm missing something? $\endgroup$ – Joaquin Brandan Aug 26 at 19:14
  • $\begingroup$ You are right, $m$ does not have a particular name. $\endgroup$ – Anatoly Aug 26 at 19:48

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