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In an equilateral triangle what is sum of distance from vertices to any arbitrary point inside the triangle? enter image description here

What is the relation between $a$ and $x + y +z$. The special condition is that the interior point cannot be considered to be a special point like centroid or circumcenter,etc.

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Vivianis theorem states that the sum of the "distances from any interior point to the sides" of an equilateral triangle equals altitude but aren't we talking about the "distances to vertices"??? –  shaurya gupta Oct 29 '13 at 18:55
    
Yes, you are right, my bad. –  Lord Soth Oct 29 '13 at 18:57
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It doesnt look like there is any "nice" formula, but if you use law of cosines on the smaller triangles and the relations we get from angles in the corners summing to $60^\circ$ you can get a messy formula relating them –  N. Owad Oct 31 '13 at 14:04
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2 Answers

up vote 1 down vote accepted

I have not even tried to simplify this, but it is a relation.

$$ \frac{a^2+x^2-y^2}{2xa} = \frac{\sqrt{3}}{2}\frac{a^2+x^2-z^2}{2xa}+\frac{1}{2} \sqrt{1- \Big(\frac{a^2+x^2-z^2}{2xa}\Big)^2} $$

Perhaps it is what you were looking for. Of course, if you require your point to be inside the equilateral, you need to have restraints on your $x,y,$ and $z$.

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It seems that the minimal value of $s=x+y+z$ is $a\sqrt{3}$ at it is attained at the centroid, and the minimal value of $s$ is $2a$ at it is attained at a vertex, and, by continuity of $s$, all intermediate values between $a\sqrt{3}$ and $2a$ are possible too.

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The values of s remain a√3 for any point that is in the interior. –  shaurya gupta Oct 30 '13 at 6:09
    
@shauryagupta It seems the following. Let the fixed point be the centroid of the triangle. Then $x=y=z=\frac 23m$, where $m$ is length of the median. But $m=a\frac{\sqrt{3}}2$. Then $s=3x=2m=a\sqrt{3}\ne 2a$. –  Alex Ravsky Oct 30 '13 at 18:44
    
yes i know that it is a√3 but what if the point is not a centroid...What if it were to be just a random point inside the triangle? –  shaurya gupta Oct 31 '13 at 13:02
    
It seems that, as I arleady wrote, all intermediate values of $s$ between $a\sqrt{3}$ and $2a$ are possible, and this value depends on the chosen point. –  Alex Ravsky Oct 31 '13 at 18:29
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