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A note on Pappus’ theorem. (English) Zbl 1027.51002
Let $$\Pi$$ be a projective plane coordinatized by a ternary ring $$(R,T)$$. For three lines $$u,v,w$$ in $$\Pi$$ the $$(u,v,w)$$-Pappus’ theorem holds if the usual Pappus configuration on the lines $$u,v$$ with points of intersection on $$w$$ closes for all suitable sets of points.
The authors prove two theorems. (1) In $$\Pi$$ the $$([0],[\infty],[1])$$-Pappus theorem holds iff the operation $$a\ast b=T(a,1,b)$$ is commutative and associative, i.e., $$(R,\ast)$$ is an abelian group.
(2) Assume that $$a\ast b=T(1,a,b)$$ then the $$((0),(1),(\infty))$$-dual-Pappus’ theorem holds iff $$\ast$$ is commutative and associative.
##### MSC:
 51A35 Non-Desarguesian affine and projective planes
##### Keywords:
projective plane; Pappus’ theorem; ternary ring
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