In an equilateral triangle what is sum of distance from vertices to a point inside the triangle?

Question

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

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.

Answer 1

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 $z$.

Answer 2

It seems that the minimal value of $s=x+y+z$ is $a\sqrt{3}$, attained at the centroid. The maximal value of $s$ is $2a$, attained at a vertex.

By continuity of $s$, all intermediate values between $a\sqrt{3}$ and $2a$ are possible too.