Pappus lines and Leisenring lines
In a Pappian plane let b and c be distinct lines meeting at 0; let \(B_ 1B_ 2B_ 3\) be distinct points of b, and let \(C_ 1C_ 2C_ 3\) be distinct points of c, all distinct from 0. Write \[ B_ 2C_ 3\cap B_ 3C_ 2=P_ 1,\quad B_ 3C_ 1\cap B_ 1C_ 3=P_ 2,\quad B_ 1C_ 2\cap B_ 2C_ 1=P_ 3; \] \[ OP_ 1\cap B_ 1C_ 1=L_ 1,\quad OP_ 2\cap B_ 2C_ 2=L_ 2,\quad OP_ 3\cap B_ 3C_ 3=L_ 3. \] Pappus' theorem states that \(P_ 1P_ 2P_ 3\) are collinear on the Pappus line of the hexagon \(B_ 1C_ 2B_ 3C_ 1B_ 2C_ 3\). The points \(L_ 1L_ 2L_ 3\) are called the Leisenring points of the hexagon. The author provides nice proofs of the following properties. The three Leisenring points are collinear on the Leisenring line of the hexagon. The six Leisenring lines associated with the six hexagons arising from two triplets of collinear points are concurrent in threes.
scientific article; zbMATH DE number 3859861