Here are some more elementary examples.
The easier cases of the AM-GM inequality follow from equalities, namely $a^2+b^2\geq 2ab$ because $a^2+b^2-2ab=(a-b)^2$ and $\frac{a^3+b^3+c^3}{3}\geq abc$ for $a,b,c\geq 0$ because $a^3+b^3+c^3-3abc=\frac{1}{2}(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2)$.
We have that for any triangle ABC the point X that minimizes $AX^2+BX^2+CX^2$ is the centroid G because of Leibniz's relation $AX^2+BX^2+CX^2=AG^2+BG^2+CG^2+3XG^2$.