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prove a triangle inequality

Albert

Well-known member
Jan 25, 2013
1,225
Triangle ABC with side lengths a,b,c please prove :

$ \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c<2\sqrt {ab}+2\sqrt {bc}+2\sqrt {ca}$
 

Albert

Well-known member
Jan 25, 2013
1,225
Triangle ABC with side lengths a,b,c please prove :

$ \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c<2\sqrt {ab}+2\sqrt {bc}+2\sqrt {ca}$
proof of left side:

$2\sqrt {ab}\leq a+b----(1)$
$2\sqrt {bc}\leq b+c----(2)$
$2\sqrt {ca}\leq c+a----(3)$
(1)+(2)+(3):$2(\sqrt {ab}+\sqrt {bc}+\sqrt {ca})\leq 2(a+b+c)$
$\therefore \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c$
 
Last edited:

Albert

Well-known member
Jan 25, 2013
1,225
proof of left side:

$2\sqrt {ab}\leq a+b----(1)$
$2\sqrt {bc}\leq b+c----(2)$
$2\sqrt {ca}\leq c+a----(3)$
(1)+(2)+(3):$2(\sqrt {ab}+\sqrt {bc}+\sqrt {ca})\leq 2(a+b+c)$
$\therefore \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c$
proof of right side:

let: $a\leq b\leq c$
$\sqrt {ab}+\sqrt {bc}\geq a+b-----(4)$
$\sqrt {bc}+\sqrt {ca}\geq a+b-----(5)$
$\sqrt {ca}+\sqrt {ab}\geq a+a-----(6)$
(4)+(5)+(6):
$2(\sqrt {ab}+\sqrt {bc}+\sqrt {ca})> 2a+2c>a+b+c$