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Background: significant parts of E. Artin, Geometric algebra, Wiley-Interscience, New York, 1957 can be seen as consequences of the statement

(1) "the inverse semigroup generated by dilations in an affine space is made by dilations and translations"

For example Menelaus theorem can be seen as a consequence of: the composition of two dilations of coefficients $\varepsilon, \mu \in (0,1)$ is a dilation of coefficient $\varepsilon \mu$.

This can be generalized to metric spaces which are metric cones with respect to any of its points. Consider a metric space $(X,d)$ with the property that for any point $x \in X$ the space and any $\varepsilon > 0$ there is a dilation $\delta^{x}_{\varepsilon}: X \rightarrow X$ which transforms the distance $d$ into $\varepsilon d$. Examples: affine metric spaces, but also Carnot groups with the Carnot-Caratheodory distance, see M Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr.Math., 144, Birkhäuser, Basel, 1996.
I am directly interested in this, because the statement (1) is true also in this general situation, see M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136

My question is: do you know of any other theorem in (metric) affine geometry which has a proof using only dilations and is not a consequence of Menelaus?

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