Sum of heights from a random point in triangle?

Question

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.

Answer 1

This is not true.

Consider O to be A, then be B. Even if you disallow O to be on the triangle, by continuity arguments, you can pick points close to A and B.