A generalisation of the nine-point circle and Euler line

Michael De Villiers

doi:10.4102/pythagoras.v0i62.112

Abstract

To most people, including some mathematics teachers, geometry is synonymous with ancient Greek geometry, especially as epitomised in Euclid's Elements of 300 BC. Sadly, many are not even aware of the significant extensions and investigations of Apollonius, Ptolemy, Pappus, and many others until about 320 AD. Even more people are completely unaware of the major developments that took place in synthetic Euclidean plane geometry from about 1750-1940, and more recently again from about 1990 onwards (stimulated in no small way by the current availability of dynamic geometry software).

Highlights

To most people, including some mathematics teachers, geometry is synonymous with ancient Greek geometry, especially as epitomised in Euclid's Elements of 300 BC
The purpose of this article is to give a brief historical background to the discovery of the Nine-point circle and the Euler line, and a simple, but possibly new generalisation and proof of the latter, that may be of interest to teachers and students
Learners who participate in the workshops and Summer School of the Mathematical Talent Search organised under the auspices of the South African Mathematical Society (SAMS) are well acquainted with this result, as are all the South African team members of the International Mathematics Olympiad (IMO)

Summary

Introduction

To most people, including some mathematics teachers, geometry is synonymous with ancient Greek geometry, especially as epitomised in Euclid's Elements of 300 BC. Learners who participate in the workshops and Summer School of the Mathematical Talent Search organised under the auspices of the South African Mathematical Society (SAMS) are well acquainted with this result, as are all the South African team members of the International Mathematics Olympiad (IMO) Homothetic polygons Another valuable result that is usually well known to successful Mathematics Olympiad contestants is the following theorem: If two polygons are homothetic (that is similar and their corresponding sides are parallel), the lines connecting corresponding vertices are concurrent at their centre of similarity (see Figure 5). It seems quite sad that such a beautiful projective geometry result has become forgotten and neglected This nine-point conic result, contains a generalisation of the Euler line as a corollary, which does not appear in any of the three references mentioned, and an internet search has provided no explicit mention of it in the mathematical literature.

Euler line generalisation

Looking back Instead of respectively using x H