Kegelschnitte und Steinersche Ovale in Fano-Ebenen. (Conics and Steiner ovals in Fano planes)
In this note there are considered conics in those projective planes, in which the Pappus theorem is valid. The author is especially interested in Fano planes. In these ones the diagonal lines of a conic play an important part. Such a line passes through the three diagonal points of an arbitrary quadrangle on a given conic. Now let be given an oval with more than seven points. It is called Steiner oval iff all diagonal lines of all conics lying on the oval have exactly one point in common (knot of the oval). Here the author firstly shows that in Pappus planes with more than seven points Steiner ovals are conics in the sense of Steiner, and secondly, that projective Fano planes are Pappus planes iff for every pentagon there exists a Steiner oval on which this pentagon is lying. Further from the last result it follows a closure theorem characterizing the Pappus planes within the set of the Fano planes.
scientific article; zbMATH DE number 4028262