MHB Triangle side terms and area inequality

AI Thread Summary
The discussion revolves around proving the inequality ab + bc + ca ≥ 4√3S for a triangle with sides a, b, c and area S. A user attempted to use the Ravi transformation but faced difficulties. Another participant suggests employing Heron's formula to express the area and reformulate the problem into an inequality involving variables x, y, and z derived from the triangle's sides. They outline the necessary steps and inequalities needed to prove the statement, referencing a mathematical text that discusses the problem. The conversation emphasizes that multiple approaches exist for tackling this inequality.
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Currently revising for my A-Level maths (UK), there is unfortunately no key in the book;
Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S

I have tried using the Ravi transformation without luck, any takers?
 
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frusciante said:
Currently revising for my A-Level maths (UK), there is unfortunately no key in the book;
Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S

I have tried using the Ravi transformation without luck, any takers?

I see that if we take the area of the triangle using the Heron's formula, which is $$S=\frac{\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4}$$, and then let $$x=a+b-c,\,y=b+c-a,\,z=a+c-b$$, the problem can be turned into a rather straightforward inequality problem...

We're asked to prove

$$\begin{align*}ab+bc+ca &\ge 4\sqrt{3}\frac{\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4}\\&=\sqrt{3(a+b+c)(a+b-c)(b+c-a)(c+a-b)}\end{align*}$$

After the substitution, expansion and simplification, we see that we need to prove:

$$(x+y)(x+z)+(x+y)(x+z)+(x+y)(x+z)\ge 4\sqrt{3xyz(x+y+z)}$$

$$x^2+y^2+z^2+3xy+3yz+3zx\ge 4\sqrt{3xyz(x+y+z)}$$

But since $$x^2+y^2+z^2\ge xy+yz+zx$$ (from the Rearrangement or AM-GM inequality), we have:

$$x^2+y^2+z^2+3xy+3yz+3zx\ge xy+yz+zx+3xy+3yz+3zx=4xy+4yz+4zx$$

I will leave it to you to prove why $$4xy+4yz+4zx \ge 4\sqrt{3xyz(x+y+z)}$$ must hold, and please take note that my proposed solution isn't the only way to tackle the problem.(Smile)
 
This problem is discussed (with solution) in the following book.

R.B. Manfrino, J.A.G. Ortega, R.V. Delgado. Inequalities: A Mathematical Olympiad Approach. 2009. p. 74, exercise 2.53.
 
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