Generalization of a geometric inequality (Q1047497)

Inequalities for medians and angle-bisectors of a triangle are discussed. The main result is following. Theorem. Let \(P\) be an arbitrary point in the plane of a triangle \(ABC\). Then \[ (PB+PC)\sin\frac{A}{2}+(PC+PA)\sin\frac{B}{2}+(PA+PB)\sin\frac{C}{2} \geq \frac{2}{3} (w_a+w_b+w_c), \] here \(w_a\), \(w_b\), \(w_c\) denote the angle-bisectors of \(ABC\). Equality holds if and only if the triangle \(ABC\) is equilateral and \(P\) is its center.