Several observations about Maneeals - a peculiar system of lines

Abstract

For an arbitrary triangle $ABC$ and an integer $n$ we define points $D_n$, $E_n$, $F_n$ on the sides $BC$, $CA$, $AB$ respectively, in such a manner that

\[\frac{|AC|^n}{{|AB|}^n} = \frac{|CD_n|}{|BD_n|}, \frac{|AB|^n}{{|BC|}^n} = \frac{|AE_n|}{|CE_n|}, \frac{|BC|^n}{{|AC|}^n} = \frac{|BF_n|}{|AF_n|},\] Cevians $AD_n, BE_n, CF_n$ are said to be the Maneeals of order $n$. In this paper we discuss some properties of the Maneeals and related objects.