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My question concerns the classical Desargues Theorem and its simplest version

The small Desargues Theorem: Let $A,B,C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$, be points such that $ab\parallel a'b'$ and $bc\parallel b'c'$. Then $ac\parallel a'c'$.

The small Desargues Theorem does not involve any Projective Geometry, so it could have been known to ancient geometers (Thales, Euclides, Proclus, Pappus etc.). Is it true or not?

Question. Is there any evidence that the small Desargues Theorem was known to ancient Greeks?

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  • $\begingroup$ Cf. this $\endgroup$ Jul 14, 2023 at 20:02
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    $\begingroup$ @J.W.Tanner Yes, I copied this question from Mathoverflow, following a suggestion of Brian Hopkins. $\endgroup$ Jul 14, 2023 at 20:28
  • $\begingroup$ I don't have an answer, but two remarks: ①What I tend to call the little (“small”?) Desargues theorem is the particular case of the Desargues theorem when the center of perspectivity lies on the axis of perspectivity (this is relevant because it holds for the projective plane over an alternative but not necessarily associative division algebra, e.g., the octonions); but what you state is a particular affine form of this little Desargues theorem (when the axis is the line at infinity). I think this is worth stating explicitly. [contd.] $\endgroup$
    – Gro-Tsen
    Jul 14, 2023 at 20:47
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    $\begingroup$ [contd.] ②Whether a theorem is known depends, of course, on exactly how you state it. Here we can state it as: if $aa'b'b$ and $bb'c'c$ are parallelograms, then so is $aa'c'c$. This is tantamount to various trivial statements about vectors or translations (e.g., if $\vec{aa'}=\vec{bb'}$ and $\vec{bb'}=\vec{cc'}$ then $\vec{aa'}=\vec{cc'}$, which of course is required to give the concept any sense), but even without these concepts it is extremely easy to see, so while I can't say whether the Greeks “knew” it, I'm confident they would have found it unproblematic. $\endgroup$
    – Gro-Tsen
    Jul 14, 2023 at 20:53
  • $\begingroup$ @Gro-Tsen Thanks for the comments. Calling this theorem the small Desargues Theorem I just follow the terminology of Robin Harnshorne from his known book "Foundations of Projective Geometry" (userpage.fu-berlin.de/aconstant/Alg2/Bib/…). This small Desargues Theorem (turned into an axiom in dimension 2) allows to introduce the addition of vectors. $\endgroup$ Jul 15, 2023 at 2:38

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