A note on an inequality involving the sides and medians in a triangle

TL;DR

This paper proves a conjectured inequality involving sides and medians of a triangle, extends it to altitudes and bisectors, and discusses related open problems in triangle geometry.

Contribution

It provides the first proof of a conjectured inequality relating sides and medians, and generalizes the inequality to other cevians like altitudes and bisectors.

Findings

The inequality holds for sides and medians of any non-degenerate triangle.

Analogous inequalities are valid when medians are replaced by altitudes or bisectors.

Open problem posed for Cevians satisfying similar inequalities.

Abstract

The main focus of the present paper is the following inequality $\left( \sqrt{bc}-a\right) m_a+ \left(\sqrt{ac}-b\right)m_b+\left(\sqrt{ab}-c\right)m_c \geq 0,$ where $a,b,c$ are the sides of a non-degenerate triangle and $m_a,m_b,m_c,$ the respective medians; which was conjectured to be true but had not been proved. We provide a proof. We also show that analogous inequality is true when the medians are replaced by the altitudes or the internal angle bisectors. Finally, we conclude with an open problem regarding the Cevians which would satisfy such inequality.