Conjecture. Let $Q$ be a finite subset of a finite-dimensional Euclidean vector space $V$ and let $p \in V$. Then $$\sum_{q \in Q} ||q - p||^2 = \sum_{q \in Q} \left(||q - \mathbb{E}[Q]||^2 + ||\mathbb{E}[Q] - p||^2 \right)$$
Note that for an individual $q \in Q$ it is not true in general that $||q - p||^2 = ||q - \mathbb{E}[Q]||^2 + ||\mathbb{E}[Q] - p||^2$. I tried many numerical examples and it always worked.
Can anyone prove or disprove this conjecture?
number the $q_i$ from $1$ to $n$ Then name $T = \sum q_i^2 = \sum q_i \cdot q_i $ which is scalar
Then $S = \sum q_i $ which is a vector with $E = S/n$
$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$
$$ \sum (p - q_i)^2 = T - 2 p \cdot S + n p^2 $$
$$ \sum ( p - \frac{1}{n}S)^2 = n p^2 - 2 p \cdot S + \frac{1}{n} S^2 $$
$$ \sum ( q_i - \frac{1}{n} S)^2 = T - \frac{1}{n} S^2 $$
and the second plus third line equals the first