Padoa's Inequality

Let in $\Delta ABC,\,$ $a=y+z,\,$ $b=z+x,\,$ $ c=x+y.$

Padoa's inequality, illustration

Then

$abc\ge 8xyz.$


Indeed, by the AM-GM inequality,

$abc=(y+z)(z+x)(x+y)\ge 2\sqrt{yz}\cdot 2\sqrt{zx}\cdot 2\sqrt{xy}=8xyz.$

Now $x,y,z\,$ can be expressed in terms of $a,b,c:$

$\displaystyle\begin{align} x &= \frac{b+c-a}{2},\\ y &= \frac{a+c-b}{2},\\ z &= \frac{a+b-c}{2}. \end{align}$

which rewrites the above inequality into

Padoa's inequality

known as Padoa's inequality after Alessandro Padoa (1868-1937). Dorin Marghidanu came up with an interesting refinement of the inequality.

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