A generalisation of the Steiner – Lehmus theorem and critical values transcendence of its parameters
Keywords:
internal n-line of a triangle, transcendental number, algebraic number field, surfaceAbstract
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 ∈ 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-lines are determined. The transcendence of positive critical values of n of the generalised Steiner – Lehmus theorem is proved.
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