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My question concerns the dependence relations between the minor theorem of Pappus which, following Heyting, I will denote by $P_9$, and (one of the) minor theorems of Desargues, $D_9$.

$P_9$ states that: "If in a hexagon two diagonal points are on the corresponding diagonals and the diagonals are concurrent, then the third diagonal point is also on its corresponding diagonal too."

$D_9$ on the other hand (is equivalent to): "Given two triangles $A_1A_2A_3$ and $B_1B_2B_3$, such that $A_i \neq B_i$, $A_iA_j \neq B_iB_j$; $A_iB_i$ $i=1,2,3$ are concurrent, $A_1 \in B_2B_3$ and $B_1 \in A_2A_3$, then the points $C_i = A_jA_k \cap B_jB_k$ ($i \neq j \neq k \neq i$) are colinear."

It is easy to show that $D_9 \Rightarrow P_9$. My question is whether the converse holds, Heyting claims this is unknown but his book "Axiomatic Projective Geometry" dates from 1980 (second ed.) so this might no longer be true, though a quick internet search failed to point me to anything useful. Does anybody know what is the status of $P_9 \Rightarrow D_9$? It would be nice considering Hessenberg's theorem showing that full Pappus implies full Desargues.

Hopefully someone can help me out, thanks in advance!

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A. Seidenberg in Pappus implies Desargues (1976) claims to correct Hessenberg's incomplete proof. From a recent review by Marchisotto (2002), I gather that Seidenberg's proof is "for real".

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  • $\begingroup$ Thanks Carlo, I am aware that Pappus does imply Desargues (I had not seen those references though and they look great!). However, I'm interested in the minor versions of the theorems and those seem to be a bit more slippery. I will check the references as soon as I'm on the university's network again. $\endgroup$
    – Juan OS
    Sep 17, 2013 at 23:24
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The converse of the Hessenberg’s theorem is not true. In the quaternionic projective plane the Desargues' theorem is true but the Pappus's is false. See historical notes in http://www.sciencedirect.com/science/article/pii/S0024379501002877# (On Pappus' configuration in non-commutative projective geometry, by Giorgio Donati).

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  • $\begingroup$ did I misunderstand the question? I thought the OP asked for a proof that Pappus implies Desargues, not the other way around? $\endgroup$ Sep 16, 2013 at 10:47
  • $\begingroup$ I'm afraid it was me who misunderstood the question. It seems OS wants an analog of jlms.oxfordjournals.org/content/s1-39/1/424.extract# (A Geometrical Proof of an Analogue of Hessenberg's Theorem, by A. D. Keedwell) with small Reidemeister configuration replaced by $D_9$. $\endgroup$ Sep 16, 2013 at 11:42
  • $\begingroup$ Zurab, the article you pointed out in the last comment seems to be, like you pointed out, closely related with what I want to find out, I'm crossing my fingers that I can get access from my uni's network. Thanks! $\endgroup$
    – Juan OS
    Sep 17, 2013 at 23:29
  • $\begingroup$ The article can be accessed also trough Library Genesis: gen.lib.rus.ec/scimag/?s=Edward+Keedwell $\endgroup$ Sep 18, 2013 at 6:01

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