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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.

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    $\begingroup$ Almost anything is a counterexample. Le't make computations easy. Our triangle is right-angled isosceles, with sides $2$, $2$, $2\sqrt{2}$. The sum of the heights is $4+2\sqrt{2}$. Let $O$ be the point where the two short legs meet. The heights are $0$, $0$, and $\sqrt{2}$. If you object that $O$ is not inside the triangle, pick a point very close to $O$ which is inside. $\endgroup$ – André Nicolas Feb 6 '12 at 20:01
  • $\begingroup$ Sorry, typo, the heights are $2$, $2$, $\sqrt{2}$, the sum is $4+\sqrt{2}$. But it is still a counterexample. $\endgroup$ – André Nicolas Feb 6 '12 at 20:25
  • $\begingroup$ It seems that you mean an arbitrary point, not a random point? $\endgroup$ – joriki Jun 7 '13 at 6:41
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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.

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