Sums of squared distances between points on an $n$-sphere I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them).  To the best of my knowledge (and my advisor's knowledge), these results are new.  However, since this is classical geometry and the proofs are not hard, I wanted to ask if anyone knows of their existence in the literature.
I have searched the literature, and I do find authors considering similar questions (e.g., what distribution of points on an $n$-sphere maximizes the sum of distances between the points) but I do not see anyone considering the sum of squared distances.
I want to emphasize that I am not asking for proofs of these results.  I have already proven them myself.
First, a bit of terminology:


*

*An $n$-sphere is the set of all points equidistance from a fixed point in $\mathbb{R}^n$.

*A set of points $\mathcal{V}$ is centrally symmetric if it is closed under the antipodal map, i.e., for every $P \in \mathcal{V}$, the point $-P$ is in $\mathcal{V}$.

*A set of points $\mathcal{V}$ is transitive if for each $P, Q \in \mathcal{V}$, there exists a symmetry of $\mathcal{V}$ such that $P$ is mapped to $Q$.


Result 1. Let $\mathcal{V}$ be a set of $V$ points on a unit $n$-sphere, and let $\mathcal{C}$ be the multiset of the lengths of all the chords between them.
Then: $$\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}(1-d^2)$$
where $d$ is the distance between the centroid of $\mathcal{V}$ and the center of the unit $n$-sphere.
This leads to a nice corollary:  The centroid of $\mathcal{V}$ coincides with the center of the $n$-sphere if and only if $\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}$.
Result 2. Let $\mathcal{V}$ be a transitive, centrally symmetric set of $V$ points on a unit $n$-sphere, and let $\mathcal{L}$ be the set of distinct lengths of all the chords between them.
Then: $$\sum_{l \, \in \, \mathcal{L}}l^2 = 2k+2$$
where $k$ is the cardinality of $\mathcal{L}$.
Question:  Has anyone seen these results in the literature?  If so, where?  (I suppose reference to any similar results would be helpful as well...)
Note:  My research was motivated by the following two facts found in the literature.  My results subsume these (if one thinks of the vertices of a regular polygon as a point configuration).


*

*For a regular polygon inscribed in a unit circle, $\sum_{c \, \in \, \mathcal{C}}c^2= V^{2}$.

*For a regular polygon inscribed in a unit circle, $\sum_{l \, \in \, \mathcal{L}}l^2 \in \mathbb{Z}$.


Fact 1 is in an unpublished paper by S. Mustonen at
https://www.survo.fi/papers/Roots2013.pdf.  Fact 2 is in a popular math book by J. Kappraff: "Beyond Measure: A
Guided Tour through Nature, Myth, and Number." (World Scientific
Publishing, River Edge, NJ, 2002). A related question was stated as
a 1923 MAA Monthly problem (Morley, F. V., Harding, A. M.: 2925. Am.
Math. Mon. 30(1), 44 (1923)).
 A: The sum of squares of chords has proven to be particularly useful in studying arrangements of lines for redundant linear encodings of data.
Suppose $n=\binom{d+1}{2}$. In this case, we may identify $\mathbb{R}^n$ with the vector space of real symmetric $d\times d$ matrices, and $\mathbb{R}\mathbf{P}^{d-1}$ with the subset of matrices that represent orthogonal projections of rank 1:
$$
\mathbb{R}\mathbf{P}^{d-1}
\cong\{xx^\top:x\in S^{d-1}\}.
$$
Notice that these matrices have unit Frobenius norm, and so they reside in the unit sphere in this space of matrices. The square of the chord length between $xx^\top$ and $yy^\top$ is given by
$$
\|xx^\top-yy^\top\|_F^2
=2-2(x^\top y)^2.
$$
As such, the sum of the squared chord lengths between pairs in $\mathcal{V}=\{x_ix_i^\top\}_{i=1}^V$ is given by
$$
\sum_{1\leq i<j\leq V}\|x_ix_i^\top-x_jx_j^\top\|_F^2
=V^2-\sum_{i=1}^V\sum_{j=1}^V(x_i^\top x_j)^2.
$$
The double sum on the right-hand side is the frame potential of $\{x_i\}_{i=1}^V$, studied extensively by Benedetto and Fickus in "Finite Normalized Tight Frames." (It's a beautiful paper!)
A: I read your paper (Sums of squared distances between points on a Unit sphere 2020 arXiv), but I must inform you that your theorem 2.1 was already established by Tom M. Apostol in a paper in Math Monthly 2003 (Sums of squares of distances in m-space) in a more general way (weighted).
A: There are two basic formulae for the moment of inertia, which I formulate for the system of unit masses: given $V$ points $p_1,\dots,p_V\in \mathbb{R}^n$, define by $J(q)=\sum |q-p_i|^2$ the moment of intertia of a point $q$. Then 
1) (Lagrange formula) $J(q)=J(z)+V|q-z|^2$, where $z$ is the centroid of our $V$ points.
2) (Jacobi formula) $J(z)=V^{-1}\sum_{i<j} |p_i-p_j|^2$.
These formulae are known under different names, I follow the book of Balk and Boltyanskiy "Geometry of masses". 
