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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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    It seems that you mean an arbitrary point, not a random point?2013-06-07

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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.