A result about Leisenring points
Let \(A_ 1\), \(A_ 2\), \(A_ 3\) and \(B_ 1\), \(B_ 2\), \(B_ 3\) be two sets of distinct collinear points on two distinct lines in a projective plane. Let \(P_ 1=A_ 2B_ 3 \cap A_ 3B_ 2\); \(P_ 2=A_ 3B_ 1 \cap A_ 1B_ 3\), \(P_ 3=A_ 1B_ 2 \cap A_ 2B_ 1\). The Pappus condition is that \(P_ 1\), \(P_ 2\), \(P_ 3\) are collinear for all choices of the \(A_ i\), \(B_ i\). Let \(O\) be the intersection of the two lines. Let \(L_ 1=OP_ 1 \cap A_ 1B_ 1\), \(L_ 2=OP_ 2 \cap A_ 2B_ 2\), \(L_ 3=OP_ 3 \cap A_ 3B_ 3\). The Leisenring condition is that \(L_ 1\), \(L_ 2\), \(L_ 3\) are collinear. The authors show that in a Moufang-Anti-Fano plane, the Leisenring condition implies the Pappus condition.
scientific article; zbMATH DE number 493213