Let $A$ be an (abstract) affine plane. We call $A$ a translation plane if the group of translations acts transitively on the set of points (Axiom 4a in Artin's book "Geometric algebra"). Desargues' "little" theorem characterizes translation planes by the following geometric assertion: Let $l_1, l_2, l_3$ be distinct parallel lines with distinct points $A_i,B_i$ on $l_i$ such that the line $A_1A_2$ is parallel to $B_1B_2$ and $A_2A_3$ is parallel to $B_2B_3$. Then $A_1A_3$ is parallel to $B_1B_3$ (see theorems 2.16, 2.17 in Artin's book).

If the lines $l_1, l_2, l_3$ are required to meet in a point, then we call the statement above just Desargues' theorem. Artin shows that Desargues' theorem is characterized by Axiom 4b: Whenever $P, Q, R$ are distinct points on a line, then there exists a dilatation sending $P$ to $Q$ and fixing $R$.

Unfortunately, Artin does not indicate whether axioms 4a and 4b are independent. In fact, in [Stevenson, Projective Planes, theorems 5.2.5 and 5.3.1] it is shown (via a lengthy detour to projective planes) that Desargues' theorem indeed implies Desargues' little theorem! Given the elementary geometric meanings, I'm wondering if there is a purely "affine" proof of this implication. I am aware that there is a nice "affine" proof of Hessenberg's theorem asserting that Pappus planes satisfying Desargues' theorem.


1 Answer 1


I just found a marvelous proof in the style I was looking for in Bennett's book "Affine and projective geometry" (see the corollary on page 60).


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