Pappus lines and Leisenring lines
type: Item
instance of: scholarly article
title: Pappus lines and Leisenring lines
zbMATH Open document ID: 0541.51004
DOI: 10.1007/BF01918135
published in: Journal of Geometry
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 51A20
zbMATH DE Number: 3859861
zbMATH Keywords: Pappian plane
zbMATH Keywords: Pappus' theorem
zbMATH Keywords: Leisenring points
zbMATH Keywords: collinear
zbMATH Keywords: Leisenring line
author: John F. Rigby
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3859861