An Inequality with Cotangents And the Circumradius

Problem

An Inequality with Cotangents And the Circumradius - problem

Solution

First recollect that

$\displaystyle \begin{align} &S=2R^2\sin A\sin B\sin C,\\ &\sum_{cycl}\cot A\cot B = 1,\\ &\sin^2\alpha=\frac{1}{1+\cot^2\alpha}. \end{align}$

Combined with the Law of Sines, these reduce the required inequality to $\displaystyle \sum_{cycl}\frac{xy}{(z+x)(z+y)}\le 1,\,$ where $x=\cot A,\,$ $y=\cot B,\,$ $z=\cot C.\,$ The latter inequality can be rewritten as

$\displaystyle \sum_{cycl}xy(x+y)\le 2xyz+\sum_{cycl}xy(x+y),$

which is obviously true, since $0\le xyz.\,$ Interestingly, equality holds for all right triangles but not for equilateral ones.

An example shows that the inequality is reversed for obtuse triangles: take, e.g., $a=\sqrt{3}\,$ and $b=c=1.$

Acknowledgment

I am grateful to Leo Giugiuc for posting a problem of his from the Gazeta Matematica at the CutTheKnotMath facebook page and later commenting with his solution.

 

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