On the relative distances between triangle centres lying on the central lines of an arbitrary plane triangle

The paper studies triangle centres and central lines of a suitable reference triangle. It derives formulas for the ratios of the distances between triangle centres along various central lines, including lines on which the Apollonius point, the centre of the Apollonius circle and the midpoint of the join of the two Brocard points lie. It identifies Eulerian lines, some already known, and some new here, these being lines involving the ratios 3 : 1 : 2 like those for HN : NG : GC′ on the Euler line itself, where H,N,G,C′ denote the orthocentre, the centre of the nine-point circle, the centroid and the circumcentre. Accurate diagrams are presented to allow a unified view to be taken of the triangle centres and of central lines considered in the paper. They motivate consideration of sets of Eulerian lines, all of which pass through the centroid G, and thereby gives rise to a number of theorems of a vectorial type in the geometry of the family of triangle centres..