Generalising a triangle inequality. Issue 555 (17th October 2018) - Retkes, Zoltan
Abstract : The main goal of this paper is to give a deeper understanding of the geometrical inequality proposed by Martin Lukarevski in [1]. In order to formulate our results we shall introduce and use the following notation throughout this paper. Let A 1 A 2 A 3 be a triangle a 1, a 2, a 3, the lengths of the sides opposite to A 1, A 2, A 3 respectively, P an arbitrary inner point of it xi, the distance of P from the side of length ai . Let r, R be the inradius and circumradius of the triangle hi, the altitude belonging to side ai, Δ the area and finally let α be a real parameter. We adopt also the use of Σ ui to refer the sum taken over the suffices i = 1, 2, 3. Now we are in the position to reformulate the original problem into a more general form namely: find bounds for Σ x α i in terms of r and R . The main results of our investigation are summarised in the following theorem.