The Euler line and where it led to

Abstract

One day, as I was struggling to construct examination questions on vectors in geometry, the following thought (1) occurred to me. (1)If the position vectors of points A, B, C are denoted by a, b, and c respectively, then the position vector of the centroid G of ΔABC is ⅓(a + b + c), but similar formulae for the circumcentre, orthocentre and incentre are not commonly known.After the thought had drifted about in my mind, it prompted a second observation (2).