Perimeter inequality

eligotas

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We have a triangle with sides a, b, c that ab+bc+ca=12
Prove that 6=<a+b+c<7
I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
 

Subhotosh Khan

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We have a triangle with sides a, b, c that ab+bc+ca=12
Prove that 6=<a+b+c<7
I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
How did you prove that (a+b+c) is greater than 6?

Please share your work - indicating exactly where you are stuck.
 

pka

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Jan 29, 2005
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We have a triangle with sides a, b, c that ab+bc+ca=12
Prove that 6=<a+b+c<7
I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
How did you prove that (a+b+c) is greater than 6?
Please share your work - indicating exactly where you are stuck.
TO eligotas, If you will answer Prof Khan's question then I will show why the upper bound.
 

eligotas

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Jan 18, 2018
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a2+b2 >= 2ab
a2+c2 >= 2ac
b2+c2 >= 2bc

a2+b2+c2 >= ab+ac+bc
a2+b2+c2+2ab+2ac+2bc >= 3ab+3ac+3bc
(a+b+c)2 >= 36
a+b+c >=6
 
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