An Inequality in Triangle with Roots and Circumradius

Problem

An Inequality in Triangle with Roots and Circumradius

Solution

Use Hölder's inequality followed by the rearrangement inequality: ,

$\displaystyle\begin{align} (a\sqrt{b}+b\sqrt{c}+c\sqrt{a})^2 &\le (a+b+c)(ab+bc+ca)\\ &\le 2s(a^2+b^2+c^2). \end{align}$

But we know that $a^2+b^2+c^2\le 9R^2.\,$ A combination of the two gives the desired result.

Acknowledgment

Dan Sitaru has kindly posted the above problem from his book Math Accent, with a solution, at the CutTheKnotMath facebook page. The solution is by Mihalcea Andrei Stefan, a grade 9 student.

 

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