Generalising a triangle inequality

Extract

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 A₁A₂A₃ be a triangle a₁, a₂, a₃, the lengths of the sides opposite to A₁, A₂, A₃ respectively, P an arbitrary inner point of it x_i, the distance of P from the side of length a_i. Let r, R be the inradius and circumradius of the triangle h_i, the altitude belonging to side a_i, Δ the area and finally let α be a real parameter. We adopt also the use of Σu_i 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 Σx^α_i in terms of r and R. The main results of our investigation are summarised in the following theorem.