TITLE: Sums of squared distances between points on an $n$-sphere
QUESTION [11 upvotes]: 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)).

REPLY [4 votes]: 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).