How do I prove this inequality: $a^2+b^2+c^2 \geq ab+bc+ac$?

Question

How do I prove that for any $a, b, c \in \mathbb{R}$ the inequality $a^2+b^2+c^2 \geq ab+bc+ac$ is true?

Answer 1

it is $$(a-b)^2+(b-c)^2+(c-a)^2\geq 0$$

Answer 2

Solution 1: rearragement inequality

Hint: WLOG, assume $a \le b \le c$. Use rearrangement inequality to conclude the desired inequality.

direct sum $a^2+b^2+c^2 \ge$ random sum $ = ab+bc+ca$, with equality holds if and only if $a = b = c$.

Solution 2: Cauchy-Schwartz inequality

Hint: set $v_1 = (a,b,c), v_2 = (b,c,a)$.

$ab+bc+ca = |\langle v_1, v_2 \rangle| \le ||v_1|| ||v_2|| = a^2+b^2+c^2$

Answer 3

Or $$a^2+b^2\geq 2ab$$ $$b^2+c^2\geq 2bc$$ and $$c^2+a^2\geq 2ca$$ Now sum all these and divide by 2.