# Perimeter inequality

#### eligotas

##### New member
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

##### Super Moderator
Staff member
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

##### Elite Member
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

##### New member
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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