# Minor theorems of Pappus and Desargues in "old school" geometry?

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!

• 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 $D_9$. Sep 16, 2013 at 11:42