Several observations about Maneeals - a peculiar system of lines

Abstract

For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner that |AC|n |CDn| |AB|n |AEn| |BC|n |BFn| = = = . |AB|n |BDn|, |BC|n |CEn|, |AC|n |AFn| Cevians ADn, BEn, CFn are said to be the Maneeals of order n. In this paper we discuss some properties of the Maneeals and related objects. 1. Introduction Given an arbitrary triangle ABC we consider certain cevians [1] defined (for given integral n) in the following way. Definition 1.1 Let ABC be a triangle and let n be an integer. There are points Dn, En, Fn on the sides BC, CA, AB respectively, satisfying |AC|n |CDn| |AB|n |AEn| |BC|n |BFn| = = = |AB|n |BDn|, |BC|n |CEn|, |AC|n |AFn|. We call the cevians ADn, BEn, CFn the order n Maneeals of the triangle ABC. It is easy to check, that medians, bisectors and symmedians of a triangle are examples of the Maneeals with integer n = 0, n = 1, n = 2 respectively. Furthermore, for the given triangle ABC, the points Dn, En, Fn (uniquely determined by the integer n), are vertices of a new triangle, which is said to be the Maneeal’s