A generalisation of the Steiner – Lehmus theorem and critical values transcendence of its parameters

Abstract

The internal n-line of a triangle is a segment from the vertex to the opposite side dividing this side into segments proportionally to the n th powers of the adjacent sides. An analogue of the Steiner – Lehmus theorem for the internal n-lines of a triangle is considered. All values n∈R for which the mentioned analogue of the Steiner – Lehmus theorem holds are found. Also all values n∈R for which there exists a non-equilateral triangle with three equal internal n-linesare determined. The transcendence of positive critical values of n of the generalised Steiner – Lehmus theorem is proved.