I've used an easy lemma for a problem about heights from a random point $O$ inside a equilateral triangle. It's easy to prove that OA'+OB'+OC'=h, where A', B' and C' are, respectively, foots of heights from $O$ to $BC$, $AC$ and $AB$.
So I was wondering if something similar could be proven in scalene triangle. Using same notation, is it true that: OA'+OB'+OC'=\frac{h_a+h_b+h_c}3
I couldn't manage to prove or counter-prove this identity.