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Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $\dim \langle a_1, a_2, a_3\rangle = 1$ (i.e. they are co-linear) and $a_1\ne a_2$, my professor defined the "simple ratio" (making a literal translation) as the unique scalar $$ (a_1a_2a_3) := \lambda \in\mathbb{K} \qquad\text{such that}\qquad \overrightarrow{a_1a_3}=\lambda\, \overrightarrow{a_1a_2}\,. $$

I haven't found any equivalent term in English sources. Is such a concept defined in other places, or does it happen to be "popular" where I live?

For example, this notation is used in "Álgebra Lineal i Geometria" by M. Castellet and I. Llerena.

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The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $\lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $\lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=\frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

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  • $\begingroup$ Please see my reference to Coxeter's book. I must admit that I am more familiar with Russian terminology, where the adjective "simple" is used instead. $\endgroup$ – GReyes Feb 17 at 9:14
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I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio $$(a_4,a_1;a_2,a_3) = \frac{a_1a_3\cdot a_4a_2}{a_1a_2\cdot a_4a_3}$$ If we choose $a_4$ to be the point at $\infty$ on the line, then $\frac{a_4a_3}{a_4a_2}=1$ and this becomes $\frac{a_1a_3}{a_1a_2}=\lambda$.

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  • $\begingroup$ The cross ratio is an invariant in Projective Geometry (simple ratio is not invariant there, and you need a ratio of ratios). Observe that the cross ratio is a ratio of simple ratios. $\endgroup$ – GReyes Feb 16 at 19:37
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Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!

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  • $\begingroup$ For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant. $\endgroup$ – GReyes Feb 16 at 19:40
  • $\begingroup$ Do you have a reference for it being named in English? $\endgroup$ – jmerry Feb 16 at 20:10
  • $\begingroup$ I was a bit surprised not to find the term "simple ratio" as often as expected. But you can check, for example, Coxeter's book (a classic) books.google.com/… $\endgroup$ – GReyes Feb 17 at 6:21

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