Some extensions of a geometric inequality generated from Ptolemy’s inequality

For convenience, throughout this Article, we shall use the standard notations for any triangle ABC, where:1) the lengths of the sides BC, CA, AB are denoted by a, b, c, respectively,2) the area of the triangle is denoted by S,3) the semiperimeter of the triangle is denoted by s,4) the circumradius and inradius of the triangle are denoted by R and r, respectively,5) the lengths of the medians corresponding to vertices A, B, C are denoted by ma, mb, mc, respectively,6) the lengths of the angle bisectors corresponding to vertices A, B, C are denoted by la, lb, lc, respectively,7) the lengths of the altitudes corresponding to vertices A, B, C are denoted by ha, hb, hc, respectively.