Notes on the Cevian nest property and the Newton property of projective planes (Q428387)

The Cevian Nest Theorem of classical projective geometry states the following: If the triangle \(A'B'C'\) is inscribed into the triangle \(ABC\), the triangle \(A''B''C''\) is inscribed into \(A'B'C'\), and both triangle pairs, \(ABC\) and \(A'B'C'\) as well as \(A'B'C'\) and \(A''B''C''\) are perspective, then \(ABC\) and \(A''B''C''\) are perspective, too. This theorem is known to hold true in Pappian projective planes.NEWLINENEWLINE The author proves the converse: If a Desarguesian projective plane satisfies the Fano axiom and the Cevian nest property, it is Pappian. (Without the assumption on the Desargue and Fano axioms, the Cevian nest property becomes meaningless.)NEWLINENEWLINEThe Newton Theorem of classical projective geometry states that the harmonic conjugates with respect to the vertices of the intersection points of a line \(l\) with a complete quadrilateral's diagonals are collinear. (If \(l\) is the ideal line of the projectively extended Euclidean plane, this simply means that the diagonal mid-points are collinear.) The Newton Theorem is true in every Pappian projective plane. Again, the author also shows necessity: If a Desarguesian projective plane satisfies the Fano axiom and the Newton property, it is Pappian.