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Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:

  1. Any distinct points $x,y\in X$ are contained in a unique line $L\in\mathcal L$;

  2. Any line $L\in \mathcal L$ contains at least three points;

  3. $X\notin \mathcal L$;

  4. For every line $L\in\mathcal L$ and point $x\in X\setminus L$ there exists a unique line $\Lambda\in\mathcal L$ such that $x\in \Lambda$ and $\Lambda\cap L=\emptyset$.

For two distinct points $x,y\in X$ of an affine space $(X,\mathcal L)$ by $\overline{xy}$ we denote the unique line containing the points $x,y$.

Two lines $A,B\in\mathcal L$ are called parallel (denoted by $A\parallel B$ ) if either $A=B$ or $A\cap B=\emptyset$.

An affine plane $(X,\mathcal L)$ is called Desarguesian if it satisfies the Affine Desargues Axiom: for every concurrent lines $A,B,C$ and points $a,a'\in A\setminus(B\cup C)$, $b,b'\in B\setminus(A\cup C)$, $c,c'\in C\setminus(A\cup B)$, the parellelity relations $\overline{ab}{\parallel} \overline{a'b'}$ and $\overline{bc}{\parallel}\overline{b'c'}$ imply $\overline{ac}{\parallel} \overline{a'c'}$.

It is well-known that every Desarguesian affine plane is isomorphic to the plane $R\times R$ over a suitable division ring $R$. Using this fact, by standard methods of analytic geometry, one can prove the following

Theorem. Let $L,L'$ be two disjoint lines in a Desarguesian affine plane and $a,b\in L$, $a',b'\in L'$ be points such that the lines $\overline{aa'}$ and $\overline{bb'}$ have a unique common point $o\notin L\cup L'$. For every points $x\in L$, $y\in \overline{ab'}$, $z\in L'$, with $\overline{xy}\parallel \overline{bb'}$ and $\overline{yz}\parallel \overline{aa'}$, the points $o,x,z$ are collinear.

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Remark. This theorem is important because it implies that a perspectivity between two parallel lines is the composition of two parallel projections and hence is an affine map between those two parallel lines.

Problem. Find a direct geometric proof of the above Theorem (which uses only geometric constructions involving only the Affine Desargues Axiom but not Linear Algebra over division rings).

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  • $\begingroup$ If was suggested to me that perhaps the way the Affine Desarguesian Axiom is stated is not quite right: there should probably be some condition on the lines $A,B,C$. $\endgroup$
    – David Roberts ♦
    Commented Sep 25, 2023 at 10:25
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    $\begingroup$ My colleague David Butler apparently had a proof of this in his honours thesis, and one of the citations was, he tells me, M J Kallaher, Affine Planes with Transitive Collineation Groups, Elsevier North Holland (1982). I can't easily find this, so it may not help. $\endgroup$
    – David Roberts ♦
    Commented Sep 25, 2023 at 11:40

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