Mathematical logic, Algebra and Number Theory

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

M. M. Vas'kovskii, M. A. Firsau, P. D. Babayeva

Belarusian State University, Minsk

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 nth powers of the adjacent sides. An analogue of the Steiner – Lehmus theorem for the internal $n$-lines of a triangle is considered. All values $n \in$ R for which the mentioned analogue of the Steiner – Lehmus theorem holds are found. Also all values $n \in$ R for which there exists a non-equilateral triangle with three equal internal $n$-lines are determined. The transcendence of positive critical values of $n$ of the generalised Steiner – Lehmus theorem is proved.

Keywords: Internal n-line of a triangle; transcendental number; algebraic number field; surface.

UDC: 514.112.3 + 511.46