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 vector algebra, however, in order to prove (ii), you have to use the coordinates and expand the brackets as follows:
\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}
So I thought (ii) is a coincidence that only works for tetrahedra. But I found an extension of this result:
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.
Now I think this is not a coincidence and there must be an elegant solution to the generalized result.
(I also noticed that the cases for dual polyhedron is similar)