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Def: Two triangles are called similar if and only if one can be scaled into the other.

AAA Theorem: If two triangles have the same angles if and only if they are similar.

How does one prove the Theorem, without relying on the trigonometric functions? I can see how to prove one direction with them (but not without them):

It is easy to say, if the angles are $a,b,c$ , the sides in the first triangle are $A_1,B_1,C_1$, and the sides in the second triangle are $A_2,B_2,C_2$. Then by law of sines,

$$\frac{A_1}{\sin a}=\frac{B_1}{\sin b}=\frac{C_1}{\sin c}$$

$$\frac{A_2}{\sin a}=\frac{B_2}{\sin b}=\frac{C_2}{\sin c}$$.

From which by division we see:

$$\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}=k$$

Where $k \in \mathbb{R}$ is some constant depending on the two triangles (ratio of two sides).

And hence $(A_1,B_1,C_1)$ is a multiple of $(A_2,B_2,C_2)$. Then up to some scaling, the sides of the triangles are the same size. So up to some scaling, the triangles are congruent by SSS.

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    $\begingroup$ Euclid Book VI, Definition 1 and Proposition 5 are probably what you want. $\endgroup$ – Chappers Jul 25 '17 at 22:26
  • $\begingroup$ On behalf of @Dan von Bose: AA could be considered a theorem since the third angle is free after knowing two of them. AAA is an axiom to combine with the congruency theorems SAS, ASA, and SSS to prove simiilarity theorems SAS and SSS. It depends. What axioms do you want to start with? $\endgroup$ – dantopa Dec 22 '18 at 1:50
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Dilate one of the triangles until one of its sides is the same length as the corresponding side of the other triangle. Dilation preserves angle measures, so they still have all their angles equal. It follows from the ASA postulate that the triangles are now congruent (and hence that the original triangles were similar).

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  • $\begingroup$ Thanks, but what about: The triangles are similar $\implies$ angles are equal. $\endgroup$ – Ahmed S. Attaalla Jul 25 '17 at 22:44
  • $\begingroup$ Actually I guess: if they triangles are similar then up to some scale $k$ they are congruent , because scaling preserves angles, (scaling by $\frac{1}{k}$) the angles must be the same. $\endgroup$ – Ahmed S. Attaalla Jul 25 '17 at 22:52
  • $\begingroup$ Depends on your approach. "Dilations preserve angles" is pretty much the first thing you prove when you start working with dilations or proportionality. Euclid does it with a slick area-based argument; you can also do unslick things like first proving it for integer dilations by induction, then using that to prove it for rational dilations, then making some kind of continuity argument. $\endgroup$ – Micah Jul 25 '17 at 22:53
  • $\begingroup$ @Micah Right; or use Thales' theorem (to prove that dilations preserve angles). $\endgroup$ – Moishe Kohan Jul 26 '17 at 2:47
  • $\begingroup$ @MoisheCohen: I don't follow. I presume that you're taking some instance of Thales' theorem, and then dilating it and argue that it's still an instance, with the same angle measures. But this probably requires some argument about off-center circles being preserved by dilation. And if you establish the general fact that dilations about different points are congruent, you don't need Thales' theorem — that's enough to show that dilations preserve angles on its own. What am I missing? $\endgroup$ – Micah Jul 26 '17 at 19:21

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