MathDB

Geometric Inequality

Source: Turkey JBMO Team Selection Test Problem 5

May 30, 2012

inequalitiesgeometryinradiusCauchy Inequalitygeometry proposed

Problem Statement

Let

a, b, c

be the side-lengths of a triangle,

r

be the inradius and

r_a, r_b, r_c

be the corresponding exradius. Show that

\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r}