• Jul 17th 2011, 09:41 AM
TheodorMunteanu
Prove that $\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} >\sqrt{a^2+ac+c^2}$,if a,b,c>0.
• Jul 17th 2011, 12:26 PM
Also sprach Zarathustra
Quote:

Originally Posted by TheodorMunteanu
Prove that $\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} >\sqrt{a^2+ac+c^2}$,if a,b,c>0.

I'm not so sure, here it goes...

I used Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia

$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} >\sqrt{a^2+ac+c^2}$

$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2} \geq 2\sqrt{\sqrt{(a^2-ab+b^2)(b^2-bc+c^2)}}\geq 2\sqrt{\sqrt{(a^2-2ab+b^2)(b^2-2bc+c^2)}}=2\sqrt{\sqrt{(a-b)^2(b-c)^2}}=2\sqrt{(a-b)(b-c)}>2\sqrt{\frac{(a+c)^2}{4}}=\sqrt{(a+c)^2}>\sqrt {a^2+ac+c^2}$
• Jul 17th 2011, 01:49 PM
Auri
You could have also substituted a, b, and c with any number greater than 0, and see if the statement returned true.

Something like.
Whops sorry, remove this-

What I meant though was substitute 1 for all three variables and see if the function is true.

$\sqrt{1^2-1*1+1^2}+\sqrt{1^2-1*1+1^2} >\sqrt{1^2+1*1+1^2}$

All though I could be completely wrong, so sorry if I am. Please correct.
• Jul 17th 2011, 01:59 PM
TheodorMunteanu
I was thinking that if we take OA=x,OB=y,OC=z with $\angle AOB=60,\angle BOC=60\Righarrow AB=\sqrt{a^2-ab+b^2},BC=\sqrt{b^2-bc+c^2},AC=\sqrt{a^2+ac+c^2}$ and using the inequality of triangle in ABC we got the conclusion.
• Jul 17th 2011, 01:59 PM
Also sprach Zarathustra
Quote:

Originally Posted by Auri
You could have also substituted a, b, and c with any number greater than 0, and see if the statement returned true.

Something like.
Whops sorry, remove this-

What I meant though was substitute 1 for all three variables and see if the function is true.

$\sqrt{1^2-1*1+1^2}+\sqrt{1^2-1*1+1^2} >\sqrt{1^2+1*1+1^2}$

All though I could be completely wrong, so sorry if I am. Please correct.

You proved that is true for a=b=c=1, but you asked to show that the equality is true for all real positive numbers a,b,c.

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Now I see that I used the condition a>b>c>0 (I assuming that it is allowed... hmmm...)
• Jul 17th 2011, 02:03 PM
TheodorMunteanu
Also try to prove that $\sqrt{x^2+y^2-\sqrt{3}xy}+\sqrt{y^2+z^2-yz} >\sqrt{x^2+z^2}$