Inequality in a triangle

Question

Let $O$ be the circumcenter and $H$ the orthocenter in a triangle with sides $a, b, c$. Is it true that $$aOA^2+bOB^2+cOC^2 \ge aHA^2 + bHB^2 + cHC^2$$ or equivalently $$(a+b+c)R^2 \ge aHA^2 + bHB^2 + cHC^2$$ where $R$ is the circumradius?

Answer 1

This is not true!

a counter example is $ a=5,b=6,c=8$, but for some $a,b,c$ satisfy it.

Edit: This is the case that $H$ is inside the circle.

it is trivial that $AH$ // $EC$,$\angle EAC=\angle EBC= \dfrac{\pi}{2}-\angle BEC =\dfrac{\pi}{2}-\angle BAC =\angle ACF \implies AE$ // $HC \implies AH=EC$

it is trivial that $EC^2=BE^2-BC^2=4R^2-a^2=AH^2$