# 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 $$f(x,y,z) = \dfrac{xyz(x+y+z)}{\Delta^2}$$.

Note by Vignesh S
2 years, 4 months ago

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@Vignesh S I'm getting the answer as

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

- 2 years, 4 months ago

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

- 2 years, 4 months ago

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.

- 2 years, 4 months ago