Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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. –  André Nicolas Feb 6 '12 at 20:01
    
Sorry, typo, the heights are $2$, $2$, $\sqrt{2}$, the sum is $4+\sqrt{2}$. But it is still a counterexample. –  André Nicolas Feb 6 '12 at 20:25
    
It seems that you mean an arbitrary point, not a random point? –  joriki Jun 7 '13 at 6:41

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.