Need for Rigor

Consider a triangle $ABC$ with area $\Delta$. Let $x = r_1 \sec \left( \dfrac A2 \right),y= r_2 \sec \left( \dfrac B2 \right)$ and $z = r_3\sec \left( \dfrac C2 \right) $, where $r_1,r_2$ and $r_3$ are the radii of the excircles of the triangle $ABC$ .

Find the infimum of the function $f(x,y,z) = \dfrac{xyz(x+y+z)}{\Delta^2}$ .

Relevant problem.

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Can @Otto Bretscher , @Pi Han Goh , @Satyajit Mohanty , @Sandeep Bhardwaj , or anybody else help me out with this. I'm not very good at inequalities, but I tried AM-GM , but its not helping much.

Vignesh S - 5 years, 2 months ago

@Vignesh S I'm getting the answer as

$3\frac { {(abc)}^{\frac{4}{3}}} {{\Delta}^{2}}$

A Former Brilliant Member - 5 years, 2 months ago

The answer is $\boxed{16}$ . Its just equilateral $\Delta$ .But I need a proof for it.

Vignesh S - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$