A pair of general triangle inequalities (Q750902)

Two `dual' triangle inequalities \[ (1/a+1/b+1/c)(R_ 1+R_ 2+R_ 3)>\begin{cases} 5 & \text{ if \(\max (A,B,C)\geq 120^\circ\)} \\ 4+2/\sqrt 3 & \text{otherwise} \end{cases} \] and \[ (a+b+c)(1/R_ 1+1/R_ 2+1/R_ 3)>2(3+2\sqrt 2) \] are proved where \(R_ 1\), \(R_ 2\), \(R_ 3\) represent the distances from P, a point of the interior or boundary of triangle ABC, to its vertices A, B and C respectively. It is also proved that the bounds cannot be improved.