Inequality Proof

[tom_smith]

I've been stuck on this question for a while now, a^2+b^2+c^2≥ ab+bc+ca

I get the basic (a-b)^2 ≥ 0,
a^2+b^2≥2ab, but I can't figure out how to get this one out.

Thanks for any help.

 

[lookagain]

I've been stuck on this question for a while now, a^2+b^2+c^2≥ ab+bc+ca

Here is a possible hint:

\(\displaystyle 2(a^2 + b^2 + c^2) \ \ge \ 2(ab + bc + ca) \)


Here is a more advanced hint:

Spoiler

\(\displaystyle a^2 \ \ \ \ \ \ \ \ \ \ \ \ b^2\)
\(\displaystyle b^2 \ \ \ \ \ \ \ \ \ \ \ \ c^2 \)
\(\displaystyle c^2 \ \ \ \ \ \ \ \ \ \ \ \ a^2 \ \ge \ 2ab + 2bc + 2ca\)

 

[skeeter]

[MATH]a^2+b^2 \ge 2ab[/MATH]
[MATH]b^2+c^2 \ge 2bc[/MATH]
[MATH]a^2+c^2 \ge 2ac[/MATH]
sum of the left sides [MATH]\ge[/MATH] sum of the right sides

 

[tom_smith]

Thanks, this helped