On Some Remarkable Points and Line Segments in Triangles

Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In the paper [Andrica D., Marinescu D.S. New interpolation inequalities to Euler's R≥2 // Forum Geometricorum. 2017. Vol. 17], the authors proved that 4/R £ 1/ra + 1/rb +1/rc £2/r. In the paper [Isaev I., Maltsev Yu., Monastyreva A. On some relations in geometry of a triangle // Journal of Classical Geometry. 2018. Vol. 4], it is given the following generalization of these inequalities: 1/ra + 1/rb +1/rc=2/R+1/r. In that paper, we find the area of the triangle OAOBOC (see Theorem 1). We prove some relations for the numbers R-ra, R-rb, R-rc, where R is the circumradius of a triangle ABC. Namely, we find the expressions 1/R-ra+1/R-rb + 1/R-rc и a/R-ra+b/R-rb + c/R-rc by means by the parameters p, R and r (see Theorem 2). We estimate these values (see Theorem 3). Finally, using the results of paper [Maltsev Yu., Monastyreva A. On some relations for a triangle // International Journal of Geometry. 2019. Vol. 8 (1)] and representing the expression of (1-cos(αβ))(1-cos(β-γ))(1-cos(α-γ)) by means of p, R, r, we prove new proof of the fundamental triangle inequality (see Corollary 2).