Über eine Äquivalenzrelation auf der Menge der (orientierten) Dreiecke einer affinen Ebene und Kennzeichnungen des papposschen Falles. (About an equivalence relation on the set of (oriented) triangles of an affine plane and characterizations of the Pappian case) (Q806036)

This article gives a characterization of pappian affine planes in terms of a property called smooth (schlicht) for equivalence relations on triangles with a common side. Let \({\mathcal A}\) be an affine plane. Consider two triangles \(\Delta_ 1=(A,B,C_ 1)\) and \(\Delta_ 2=(A,B,C_ 2)\) with a common side AB. The triangles \(\Delta_ 1\) and \(\Delta_ 2\) are primary-equivalent if the points \(C_ 1\) and \(C_ 2\) lie on a line parallel to the line AB. An arbitrary equivalence relation \(\sim\) on the set of triangles in \({\mathcal A}\) is smooth, and the plane \({\mathcal A}\) is smooth, if \((A,B,C_ 1)\sim (A,B,C_ 2)\) implies the two triangles are primary-equivalent. The author proves that an affine plane \({\mathcal A}\) is pappian if and only if it is smooth with respect to some equivalence relation on the set of triangles in \({\mathcal A}\). More, however, is accomplished. The author shows that the Pappus configuration theorem is equivalent to five other conditions; these include an interesting configuration theorem and two statements about an area mapping on the set of triangles.