This is a variant of the result discussed in this link: On a constant associated to equilateral triangle and its generalization.
Consider any regular polygon and an arbitrary point, $P$, on an arbitrary circle with center at the centroid of the polygon. The sum of the square distances from $P$ to the sides of the polygon is a constant.
This can be extended to a tetrahedron. Consider a tetrahedron $ABCD$ and an arbitrary point, $P$, on any sphere with center at the centroid of the tetrahedron. The sum of the square distances from $P$ to the planes containing the faces of the tetrahedron is a constant. I suspect this can be generalized to any regular polyhedra.
These two similar results are probably particular cases of a more general theorem, so my question is: which whould be this general theorem for which these two results are just particular cases?
Best Answer
Let $A_1$, $A_2$, $A_3$, … , $A_n$ be a set of points, and let $G=\displaystyle\frac{A_1+A_2+\ldots+A_n}{n}$.
Now take any circle centered at $G$. (If you're working in 3 dimensions, take a sphere centered at $G$ instead.) For any point $P$ on this circle, the value $A_1P^2+A_2P^2+\ldots+A_nP^2$ is constant.
To prove this, first note that, for any point $X$, we have $\vec{A_1X}+\vec{A_2X}+\ldots+\vec{A_nX}=n\vec{XG}$.
From this, you can conclude that $\vec{A_1G}+\vec{A_2G}+\ldots+\vec{A_nG}=0$.
So, for any point $X$, we have the following:
$$A_1X^2+A_2X^2+\ldots+A_nX^2=\left|\vec{A_1X}\right|^2+\left|\vec{A_2X}\right|^2+\ldots+\left|\vec{A_nX}\right|^2$$ $$=\left|\vec{A_1G}+\vec{XG}\right|^2+\left|\vec{A_2G}+\vec{XG}\right|^2+\ldots+\left|\vec{A_nG}+\vec{XG}\right|^2$$
$$=n\cdot\left|\vec{XG}\right|^2+2\cdot\vec{XG}\cdot\left(\vec{A_1G}+\vec{A_2G}+\ldots+\vec{A_nG}\right)+\left|\vec{A_1G}\right|^2+\left|\vec{A_2G}\right|^2+\ldots+\left|\vec{A_nG}\right|^2$$ $$=n\left|\vec{XG}\right|^2+\left|\vec{A_1G}\right|^2+\left|\vec{A_2G}\right|^2+\ldots+\left|\vec{A_nG}\right|^2$$ $$=n\cdot {XG}^2+A_1G^2+A_2G^2+\ldots+A_nG^2$$
So, when $XG$ is constant, $A_1X^2+A_2X^2+\ldots+A_nX^2$ is constant.
Sources: http://www.cut-the-knot.org/triangle/medians.shtml
Side note if you like physics:
Imagine $n$ Hookean springs, all with the same spring constant. The first spring connects $A_1$ to $X$, the second spring connects $A_2$ to $X$, and etc. In this setup, points $A_1$ through $A_n$ are fixed, but point $X$ is free to move. The potential energy of this setup is proportional to $A_1X^2+A_2X^2+\ldots+A_nX^2$. Since nature likes to minimize potential energy, this setup will come to rest when $A_1X^2+A_2X^2+\ldots+A_nX^2$ is at it's minimum. This setup will also come to rest when all the forces sum to $0$. At any point $X$, the sum of the forces will be $k\cdot\left(\vec{A_1}+\vec{A_2}+\ldots+\vec{A_n}\right)=k\cdot n\cdot XG$, where $k$ is the spring constant of each spring. This vector is sort of the opposite of a gradient vector, in the sense that it points to the direction of greatest decrease. If point $X$ moves perpendicular to this vector, there will be no change in the value of $A_1X^2+A_2X^2+\ldots+A_nX^2$. Ergo, you can move $X$ along a circle centered at $G$, and the value $A_1X^2+A_2X^2+\ldots+A_nX^2$ will remain constant.