# A constant related to distances from a point on the circumsphere of regular polyhedron to the vertices

I was interested in problem 3 of STEP3 2013(I have shortened the original description):

The four vertices $$P_{i}(i=1,2,3,4)$$ of a regular tetrahedron lie on the surface of a sphere with center at $$O$$. Let $$X$$ be any point on the surface of the sphere, and let $$X P_{i}$$ denote the distance between X and$$P_{i}$$.
Show that (i) $$\sum_{i=1}^{4}\left(X P_{i}\right)^{2}$$ (ii) $$\sum_{i=1}^{4}\left(X P_{i}\right)^{4}$$ are both independent of the position of X.

(i) could be easily proven using vectors.

To prove (ii), I have to use the coordinates and expanding:

\begin{aligned} &\sum_{i=1}^{4}\left(X P_{i}\right)^{4}=16R^2+4\left(z^{2}+\left(\frac{2 \sqrt{2}}{3} x-\frac{1}{3} z\right)^{2}+\left(-\frac{\sqrt{2}}{3} x+\frac{\sqrt{2}}{\sqrt{3}} y-\frac{1}{3} z\right)^{2}+\left(-\frac{\sqrt{2}}{3} x-\frac{\sqrt{2}}{\sqrt{3}} y-\frac{1}{3} z\right)^{2}\right)\\ &=16R^2+4\left(\frac{4}{3} x^{2}+\frac{4}{3} y^{2}+\frac{4}{3} z^{2}\right)=\frac{64}{3} R^2 \end{aligned}

I find some similar results:

$$X$$ is a point on the circumsphere of a regular polyhedron of $$n$$ faces with edge length 1, and let $$S(k)$$ denote the sum of the $$2k$$ th power of the distances between $$X$$ and the vertices.

n=4,S(1)=3,S(2)=3,S(3) is not constant.

n=6,S(1)=12,S(2)=24,S(3)=54,but S(4) is not constant.

n=8,S(1)=6, S(2)=8,S(3)=12, but S(4) is not constant.

n=12,S(1)=$$45+15\sqrt5$$,S(2)=$$210+90\sqrt5$$,$$S(3)=1215+540\sqrt5$$,
S(4)$$= 7614 + 3402\sqrt5$$,S(5)$$=49815 + 22275\sqrt5$$,but S(6) is not constant.

n=20,S(1)=$$15+3\sqrt5$$,S(2)=$$30+10\sqrt5$$,S(3)=$$75+30\sqrt5$$,S(4)=$$210+90\sqrt5$$,
S(5)=$$625+275\sqrt5$$, but S(6) is not constant.
(I also noticed that the cases for dual polyhedron is similar)

Is there a better proof of these results than just expanding expressions?

• There is nowhere to generalize to: we don't have more regular polyhedra. (Those in higher dimensions can also be checked one by one.) That being said, I hear the spirits of irreducible representations of symmetry groups knocking at our door. Commented Sep 20, 2020 at 20:07