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 m a , m b , m c , respectively, 6) the lengths of the angle bisectors corresponding to vertices A , B , C are denoted by l a , l b , l c , respectively, 7) the lengths of the altitudes corresponding to vertices A , B , C are denoted by h a , h b , h c , respectively.