Inequalities

If $a,b$ and $c$ are sides of a triangle $ABC$ with area $X$ , prove that $ab + bc + ca \geq 4 \sqrt3 X$ .

#Algebra

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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

By rearrangement inequality, $a^2 + b^2 + c^2 \geq ab + ac + bc$ , then apply Weitzenböck's inequality, the result follows.

Pi Han Goh - 4 years, 8 months ago

ok but i have a doubt we know that a^2+b^2+c^2>=ab+bc+ca and a^2+b^2+c^2>=4(square root of 3)(X) but how u got the relation between these two since both are less than a^2+b^2+c^2

Subham Subian - 4 years, 8 months ago

Are you sure you got your question right? After reading your question for the second time, I'm pretty sure your inequality sign should be reversed.

Pi Han Goh - 4 years, 8 months ago

@Pi Han Goh – it was in a book(challenge and thrills of pre college mathematics) but i write the question write

Subham Subian - 4 years, 8 months ago

@Pi Han Goh – The correct question is to prove;

$\large\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }\ge 4\sqrt { 3 } X$

Priyanshu Mishra - 4 years, 8 months ago

@Priyanshu Mishra – It is evident from Paul-Erdos Inequality

Priyanshu Mishra - 4 years, 8 months ago

Counterexample: Consider the 3-4-5 triangle for which we have $(a,b,c)=(3,4,5)$ and hence $ab+bc+ca=12+20+15=47$ whereas the area $X=\frac{3\times 4}2=6$ from which we have $4\sqrt 3X=24\sqrt 3\approx 41.569$ , so we have $ab+bc+ca\gt 4\sqrt 3X$

However, when the inequality sign is reversed, the claim is correct and the proof for that is given by Pi Han Goh in his comment.

Prasun Biswas - 4 years, 8 months ago

Thanks! I've reversed the sign.

Calvin Lin Staff - 4 years, 7 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}`	$\boxed{123}$