An upper bound for a sequence of cevian inequalities (Q1083694)

The altitudes, Gergonne cevians, internal angle bisectors, medians and Nagel cevians of a triangle satisfy the sequence of inequalities \(\Sigma h_ a\leq \Sigma g_ a\leq \Sigma w_ a\leq \Sigma m_ a\leq \Sigma n_ a.\) The author shows that \(\Sigma n_ a\leq 14R-19r,\) where R and r are respectively the circumradius and inradius of the triangle, with equality throughout the sequence when the triangle is equilateral.